Grade 12 / Limits of numerical functions 100 exercises (including 99 corrected)

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-5-102∞−∞-3-54-4,5xVariationdef −∞-21¸∞∞∞−∞-10∞xVariationdef ijC 1. General reminders E.3307 Multiple-choice questionnaire: For each question, only one of the three answers is correct. A correct answer earns 0.5 point a wrong answer deducts 0.25 point ; no answer earns 0 point. If the total points are nega-tive, the overall mark awarded is 0 . Let f be a function defined and derivable on the interval − 5 ; + ∞ whose table of variations is given below : The representative curve of f is C : 1 On the interval − 5 ; + ∞ , the equation f ( x )= − 2 : a admits only one solution b admits two solutions c admits four solutions. 2 On the interval − 5 ; + ∞ , the function f : a admits for minimum the value − 5 ; b admits for maximium the value 4 ; c admits two maximuns. 3 We know that the equation of the tangent at C at the point of abscissa 2 is : a y = 4 b y = 4( x − 2) c x = 4 E.3311 Consider a function f defined and derivable on each of the intervals −∞ ; − 2 and − 2 ; + ∞ . The function f admits the varition table below : où ¸ is the real number strictly greater than 1 such that f ( ¸ )=0 . The graphical representation of the function f in an orthonor- mal reference frame O ; −→ i ; −→ j is called C . Say for each of the following five statements whether it is ˇ true ı or whether it is ˇ false ı or if ˇ we cannot conclure ı. No justification is required. The scale is as follows : 0.5 point per correct answer; 0.25 point per wrong answer; 0 point for no answer. There will be no negative overall score. 1 The equation f ( x )=1 admits exactly two solutions. 2 f ( x ) 0 for all x ∈ − 5 ; − 2 . 3 If − 2 <x< 1 and x<x then f ( x ) <f ( x ) . 4 f ( x ) 0 for any x ∈ −∞ ; − 2 . E.5018 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider a function f derivable on the interval − 3 ; 2 . The following information is available: f (0)= − 1 the derivative f of the function f admits the representa-tive curve C below. For each of the following statements, say whether it is true or false and justify the answer. 1 For any real x from the interval − 3 ; − 1 , f ( x ) 0 . 2 The function f is increasing on the interval − 1 ; 2 . 3 For any real x in the interval − 3 ; 2 : f ( x ) − 1 . 4 Let C be the representative curve of the function f . The tangent to the curve C at the point of abscissa 0 passes through the point with coordinates (1 ; 0) . 2. Reminders: second-degree polynomials E.3343 Each of these polynomials admits at least one root from the following set : − 2 ; − 1 ; 1 ; 2 Use this information to perform " quickly "factorization of each of the following polynomials: a x 2 + 2 x − 8 b 2 x 2 − 4 x − 6 c x 2 + x − 6 d 3 x 2 − 4 x + 1 e x 3 + x 2 − 2 x f 5 x 2 + 3 x − 2 https://chingmath.fr chapExoCorrec/3307 sacados/3307 -5-102∞−∞-3-54-4,5xVariationdef chapExoCorrec/3311 sacados/3311 −∞-21¸∞∞∞−∞-10∞xVariationdef chapExoCorrec/5018 sacados/5018 Metropole Juin 2012 4 points ijC chapExoCorrec/3343 sacados/3343
-7-6-5-4-3-2-12I-3-2-12JOCfCg -2-123456I-2-12JOCf E.3344 Draw up the table of variations of each of the following second-degree polynomials: a x 2 − 5 x + 1 b − 3 x 2 + x − 1 c 2( x + 1)(2 x − 1) d (2 − x )(4 + x ) E.3860 Solve the following equations : a 3 x 2 + 4 x + 1 = 0 b 3 x 2 − 4 x + 2 = 0 c − x 2 + 2 x + 3 = 0 d 2 x 2 − 4 x + 2 = 0 e − 3 x 2 + 3 x + 3 = 0 f − x 2 + 4 x + 3 = 0 E.4986 Factor, if possible, the second-degree polynomials below : a 3 x 2 + 4 x + 1 b − 3 x 2 + 4 x − 1 c − 4 x 2 + 5 x d x 2 + 2 x − 1 e − x 2 + 4 x + 1 f 3 x 2 − 4 x + 2 E.4987 In the plane provided with a reference frame O ; I ; J , consider the curves C f and C g representative of the functions f and g defined by: f ( x ) = x 2 + x − 2 ; g ( x ) = − 1 4 · x 2 − 2 · x − 2 The questions below must be answered algebraically: 1 Determine the antecedents of 0 by the functions f and g . 2 a Determine the coordinates of the intersection points of the curves C f and C g . b Deduce the relative position of these two curves. 3. Reminders: derivatives E.3304 Consider the function f whose image of x ∈ R is defined by the following second-degree polynomial: f ( x ) = x 3 − 2 x 2 + x − 3 1 a Determine the expression of the derivative f of the function f . b Determine the sign table of the function f on R . 2 Deduce the table of variations of the function f (we will complete the table of variations using approximate val-ues) . 3 Using the table of variations, justify that the equation f ( x ) = 0 admits a single solution. E.4996 Consider the function f defined by: f : x ↦−→ − x 3 − 3 x 2 − 2 x 1 Determine the zeros of the function f . 2 a Give the expression of the function f derived from the function f . b Draw up the table of variations of the function f on R . (we’ll omit the values in the table of variations) E.4988 Consider the function f defined on R by the relation: f ( x ) = 1 3 · x 3 − 3 2 · x 2 + x + 1 The curve C f representative of the function f in the reference frame O ; I ; J below is given : Note (Δ) the tangent to the curve C f at the point of abscissa 3 . 1 a Determine the coordinates of the point on the curve C f having abscissa 3 . b Determine the slope of the line (Δ) . c Draw the line (Δ) in the reference frame. 2 Determine, algebraically, the slope-intercept formof the line (Δ) . https://chingmath.fr chapExoCorrec/3344 sacados/3344 chapExoCorrec/3860 sacados/3860 chapExoCorrec/4986 sacados/4986 chapExoCorrec/4987 sacados/4987 -7-6-5-4-3-2-12I-3-2-12JOCfCg chapExoCorrec/3304 sacados/3304 chapExoCorrec/4996 sacados/4996 chapExoCorrec/4988 sacados/4988 -2-123456I-2-12JOCf
IJO -12-8-44812I-12-8-44812JOCf E.3310 Let be the equation ( E ): 1 x = x − 2 où where the unknown is a real from the interval 0 ; + ∞ . 1 A student plotted on his calculator the hyperbola with equation y = 1 x and the straight line with equation y = x − 2 . In view of the graph above obtained on the screen of his calculator, how many solu-tions does the equation ( E ) seem to admit on 0 ; + ∞ 2 A second student considers the function g defined on 0 ; + ∞ by: g ( x ) = x − 2 − 1 x a Let g be the derivative function of g . Calculate g ( x ) . Show that g is strictly increasing on 0 ; + ∞ . b Determine the images, by the function g , of the num-bers 1 and 4 . Conjecture the number of solutions of the equation ( E ) . 3 A third student says : ˇI can solve the equation ( E ) al-gébriquementı. Justify, by solving the equation ( E ) , that this third student is right. 4. Introduction to function limits E.4989 Consider the function f defined by: f : x ↦−→ x + 1 x 1 Give the definition set of the function f . 2 a By mental calculations, Complete the table of values below : x 1 10 100 1000 f ( x ) b What can we say about the value of ˇ f ( x ) ı when ˇ x ı grows enormously? 3 a Using mental calculations, complete the table of val-ues below : x 1 10 − 1 10 − 2 10 − 3 f ( x ) b What can we say about the value of ˇ f ( x ) ı when ˇ x ı remains positive but becoming smaller and smaller? Below is given the curve C f representative of the function f in a O ; I ; J orthonormal reference frame : 4 Interpret graphically the results of question 3 . E.2289 Consider the function f whose image of a number x is defined by the relation: f ( x ) = sin 1 x 1 Determine the definition set of the function f . 2 a Copy and complete the value table below : x 1 2 ı 1 4 ı 1 40 ı 1 2 72 ı f ( x ) b Copy and complete the value table below : x 2 ı 2 5 ı 2 41 ı 2 (2 101 +1) ı f ( x ) c Can we talk about a limit for this function when x approaches 0? 3 Draw the representative curve C f of this function, then zoom in on the reference point. What do you see? https://chingmath.fr chapExoCorrec/3310 sacados/3310 IJO chapExoCorrec/4989 sacados/4989 -12-8-44812I-12-8-44812JOCf chapExoCorrec/2289 sacados/2289
-4-2024-224Cf -4-2024-224Cg -4-2024-224Cf -4-2024-224Cg -8-4481216I-12-8-44812JOCf -22I-224JO E.5002 The plane is provided with a reference frame O ; I ; J orthonormal in which are represented the curves C f and C g representative of the functions f and g : Graphically, give, if possible, the value of the following limits: a lim x ↦→ + ∞ f ( x ) b lim x ↦→−∞ f ( x ) c lim x ↦→ 1 + f ( x ) d lim x ↦→−∞ g ( x ) e lim x ↦→ + ∞ g ( x ) f lim x ↦→− 1 + g ( x ) E.647 The plane is provided with a reference frame O ; I ; J orthonormal in which are represented the curves C f and C g representative of the functions f and g : Graphically, give, if possible, the value of the following limits: a lim x ↦→−∞ f ( x ) b lim x ↦→ + ∞ f ( x ) c lim x ↦→ 1 − f ( x ) d lim x ↦→−∞ g ( x ) e lim x ↦→ + ∞ g ( x ) f lim x ↦→− 1 − g ( x ) E.4990 Consider the function f defined by: f : x ↦−→ x + 2 x − 4 1 Give the definition set of the function f . 2 Determine the value of the reals a and b realizing the following identity for any x ∈D f : f ( x ) = a + b x − 4 3 Using the expression obtained in the previous question : a What can be said about the value of f ( x ) when the value of x grows indefinitely? b What can we say about the value of f ( x ) when x be-longs to 4 ; 5 and its value gets closer and closer to 4 ? c What can we say about the value of f ( x ) when x be-longs to 3 ; 4 and its value gets closer and closer to 4 ? Below is given the curve C f representative of the function f in a O ; I ; J orthonormal reference frame : 4 Interpret graphically the results previously obtained. E.2308 Consider the function f defined on R by the relation: f ( x ) = 2 5 x 2 − 1 Note C f the rep-resentative curve of the function f in the O ; I ; J orthonormed frame given opposite : The aim of the ex-ercise is to deter-mine the reduced equation of the tan-gent, noted ( T ) , to the curve C f at the point of abscissa 1 . 1 a Show that for h a non-zero real number, we have : f (1+ h ) − f (1) h = 4 5 + 2 5 · h . b Deduce the value of the derivative number f (1) of the function f in 1 . 2 a Justify that the expression of the tangent ( T ) is of the form : y = 4 5 · x + b b Give the coordinates of the point belonging to the curve C f having abscissa 1 . c Determine the reduced expression of the tangent ( T ) . 3 Draw the tangent ( T ) in the above reference frame. E.6751 Graphically and using a calculator, de-termine the following limits: a lim x ↦→ + ∞ x − 1 3 − x b lim x ↦→ 1 − √ 1 − x x − 1 c lim x ↦→− 2 + − 2 · x 2 − 2 · x + 4 2 · x + 4 d lim x ↦→ 0 − x 2 + x · 1 − 1 x https://chingmath.fr chapExoCorrec/5002 sacados/5002 -4-2024-224Cf -4-2024-224Cg chapExoCorrec/647 sacados/647 -4-2024-224Cf -4-2024-224Cg chapExoCorrec/4990 sacados/4990 -8-4481216I-12-8-44812JOCf chapExoCorrec/2308 sacados/2308 -22I-224JO chapExoCorrec/6751 sacados/6751
5. Limits without indeterminate shapes E.5704 Determine the limits below : a lim x ↦→ 1 + x 2 − 3 x + 5 x − 1 b lim x ↦→ 1 + 5 − x 1 − x c lim x ↦→ 0 − x 2 + x + 1 x d lim x ↦→ 0 + x − 1 − x 3 E.5703 Determine the value of the following limits: a lim x ↦→ + ∞ x + 1 x b lim x ↦→ + ∞ x + 1 1 + 1 x c lim x ↦→−∞ x 2 − x + 2 d lim x ↦→−∞ x + 2 1 − x e lim x ↦→ 0 − x + 1 x f lim x ↦→ 1 + x − 1 x + x x − 1 6. Boundaries with indeterminate shapes E.3336 Without determining the limits, specify which have an indeterminate form : a lim x ↦→ + ∞ 1 + 1 x x 2 + 2 · x − 1 b lim x ↦→ 0 + x 2 + 2 · x x 3 − x 2 c lim x ↦→ 2 + x 2 − x − 2 · x − 2 d lim x ↦→ 0 + x − 1 x 2 + x e lim x ↦→−∞ x 3 − 2 x 2 + 1 f lim x ↦→ + ∞ x 3 − 2 x 2 + 1 E.6787 Without determining the limits, specify which have an indeterminate form : a lim x ↦→ + ∞ x + x b lim x ↦→ + ∞ x − x c lim x ↦→ 0 + x − 1 x d lim x ↦→ + ∞ x − 1 x e lim x ↦→ 0 + x + 1 x f lim x ↦→ + ∞ x + 1 x E.4993 Determine the value of the following limits: a lim x ↦→ + ∞ 2 · x 2 + 3 · x b lim x ↦→ + ∞ − 3 · x 3 + 2 x c lim x ↦→−∞ 5 · x 2 − 3 · x 3 d lim x ↦→ + ∞ x 2 + 2 · x − ( x + 2) 2 e lim x ↦→−∞ x 2 + 3 · x f lim x ↦→−∞ x 3 − 2 · x 2 E.5705 Determine the value of the limits below : 7. Limits of rational fractions at infinity E.4992 1 Consider the function f defined by: f : x ↦−→ 3 x 2 + 5 x 3 x 3 + 4 x + 1 a Establish the following equality: 3 x 2 + 5 x 3 x 3 + 4 x + 1 = 3 + 5 x x · 3 + 4 x 2 + 1 x 3 b Deduct the value of the limit: lim x ↦→ + ∞ f ( x ) . 2 Consider the function g defined by: g : x ↦−→ 4 x 3 + 2 x + 1 2 x 3 − 2 x 2 By analogous reasoning to the previous question, estab-lish the following equality: lim x ↦→ + ∞ g ( x ) = 2 E.607 Find the values of the following limits: a lim x ↦→ + ∞ x 2 + 2 x + 1 b lim x ↦→−∞ x − 3 2 · x 2 − 3 c lim x ↦→−∞ 4 · x 2 − 3 · x + 2 − 3 · x 2 d lim x ↦→−∞ x 5 + x 4 x 3 − x e lim x ↦→ + ∞ 5 · x 4 − 2 · x 3 3 · x 2 − 2 f lim x ↦→ + ∞ 2 · x 10 + x 6 · x 10 − 2 · x 3 8. Limits of rational fractions in 0 E.5013 Consider the function f whose image of a number x is defined by the relation: f ( x )= 2 · x 3 − x x 3 +2 · x 2 1 Establish the following identities: https://chingmath.fr chapExoCorrec/5704 sacados/5704 chapExoCorrec/5703 sacados/5703 chapExoCorrec/3336 sacados/3336 chapExoCorrec/6787 sacados/6787 chapExoCorrec/4993 sacados/4993 chapExoCorrec/5705 sacados/5705 chapExoCorrec/4992 sacados/4992 chapExoCorrec/607 sacados/607 chapExoCorrec/5013 sacados/5013
f ( x ) = 2 − 1 x 2 1 + 2 x = 2 · x 2 − 1 x · ( x + 2) 2 Determine the value of the following limits: lim x ↦→ + ∞ f ( x ) ; lim x ↦→ 0 + f ( x ) E.4994 Determine the value of the following limits: a lim x ↦→ 0 + 1 x b lim x ↦→ 0 − 1 x c lim x ↦→ 0 − 3 x 2 d lim x ↦→ 0 + x 3 + 2 · x x 2 + x e lim x ↦→ 0 + − 2 · x 2 x 4 + x 3 f lim x ↦→ 0 − − 2 · x 2 x 4 + x 3 9. Limits of rational fractions with factoring E.635 Determine the value of the follow-ing limits: a lim x ↦→ 2 + 2 − x 2 · x 2 − x − 6 b lim x ↦→ 3 − x − 3 2 · x 2 − 15 · x + 27 c lim x ↦→ 1 − − 3 · x 2 + 7 · x − 4 ( x − 1) 2 d lim x ↦→− 2 + 3 · x 2 + 5 · x − 2 x 2 + 7 · x + 10 10. Limits of rational fractions with sign table E.5014 Consider the function f defined by the relation: f ( x ) = x 2 + 3 · x 2 · x 2 − 10 · x + 12 1 Establish the following identities: f ( x ) = 1 + 3 x 2 − 10 x + 12 x 2 = x · ( x + 3) ( x − 2)(2 · x − 6) 2 Draw up the sign table for ( x − 2)(2 x − 6) . 3 Deduce the value of the following limits: a lim x ↦→−∞ f ( x ) b lim x ↦→ + ∞ f ( x ) c lim x ↦→ 2 − f ( x ) d lim x ↦→ 2 + f ( x ) e lim x ↦→ 3 − f ( x ) f lim x ↦→ 3 + f ( x ) E.617 Determine the value of each of the following limits: a lim x ↦→ 2 + 1 2 · x 2 − 5 · x + 2 b lim x ↦→ 2 + 1 2 · x 2 − 12 · x + 16 E.8656 Determine the value of each of the fol-lowing limits: a lim x ↦→− 1 − x + 4 3 x 2 − x − 4 b lim x ↦→ 1 − x 2 + x − 3 3 · x 2 − 7 · x + 4 11. Limits of rational fractions E.6801 Consider the polynomial P =3 · x 2 − x − 2 . 1 Factor the polynomial P , leaving a record of your ap-proach. 2 Determine the value of the following two limits: a lim x ↦→ 1 + 2 · x − 3 3 · x 2 − x − 2 b lim x ↦→ 1 + 2 · x − 2 3 · x 2 − x − 2 E.3337 Determine the value of each of the following limits: E.2808 Each of the limits below has an inde-terminate form. Demonstrate the expected result by showing the correct algebraic transformation : a lim x ↦→ 0 − 1 x 2 + 1 x = + ∞ b lim x ↦→ + ∞ 5 x 2 + 3 2 x 2 − 3 x + 1 = 5 2 c lim x ↦→ 3 x 2 − 5 x + 6 x − 3 = 1 E.3367 Determine the value of each of the fol-lowing limits: a lim x ↦→ + ∞ 3 x 6 + 2 x 3 2 x 8 − x 3 b lim x ↦→ 0 3 x 5 − x 3 x 4 + x 3 c lim x ↦→ 3 + 1 − 2 x 2 + 4 x + 6 d lim x ↦→ 2 + x 2 − 3 x + 2 − 3 x 2 + 5 x + 2 E.5011 Determine the value of the following limits: a lim x ↦→ + ∞ 2 · x 3 + x − 2 − x 2 − 2 b lim x ↦→ 0 + 3 · x 3 − 2 · x 2 x 4 + x 3 c lim x ↦→ 3 + x − 6 2 x 2 − 15 x + 27 d lim x ↦→− 1 + 2 x 2 + 5 x + 3 − x 2 + x + 2 E.5749 Determine the value of the following limits: a lim x ↦→−∞ x 4 − 3 x 2 x 2 − 4 x 4 b lim x ↦→− 2 + x − 1 2 x 2 + 5 x + 2 c lim x ↦→ 2 + 4 − 2 x 3 x 2 − 4 x − 4 https://chingmath.fr chapExoCorrec/4994 sacados/4994 chapExoCorrec/635 sacados/635 Ne fait intervenir que des simplifications chapExoCorrec/5014 sacados/5014 chapExoCorrec/617 sacados/617 Ne fait intervenir que des tableaux de signes chapExoCorrec/8656 sacados/8656 chapExoCorrec/6801 sacados/6801 chapExoCorrec/3337 sacados/3337 Fait intervenir des tableaux de signes et des simplifications chapExoCorrec/2808 sacados/2808 chapExoCorrec/3367 sacados/3367 chapExoCorrec/5011 sacados/5011 chapExoCorrec/5749 sacados/5749
E.5026 Determine the value of the following limits: a lim x ↦→ 0 + 1 x − x 2 b lim x ↦→− 1 + x 2 + 6 x + 3 2 · x 2 + x − 1 c lim x ↦→ + ∞ 3 x 2 + 14 x − 4 x + 5 − 3 x + 1 E.3641 Determine the value of each of the fol-lowing limits: a lim x ↦→ + ∞ 5 x 2 + 3 x − 1 3 x 3 + 2 x b lim x ↦→ 0 − 3 x 4 − 2 x x 3 + x 2 c lim x ↦→ 2 + x 2 − 4 x + 4 − 2 x 2 + 10 x − 12 d lim x ↦→− 1 + 1 2 x 2 + x − 1 E.2960 Each of the limits below represents an indeterminate form ; perform the appropriate algebraic trans-formations to determine each of its limits: a lim x ↦→ 1 + 1 x − 1 − 1 x 2 + x − 2 b lim x ↦→−∞ 5 x 2 − 3 x + 1 2 x 2 + x − 2 12. Limits and radicals: conjugated expression E.4991 1 Establish the following algebraic equality for x ∈ R ∗ : x x 2 + 1 − x + 1 = x 2 + 1 + x + 1 x − 1 2 Deduce the value of the limit: lim x ↦→ 0 + x x 2 + 1 − x + 1 = − 2 E.3338 Determine the value of each of the fol-lowing limits: a lim x ↦→ 1 + 1 − x x + 3 − 2 b lim x ↦→− 1 + x + 1 x 2 + 4 x + 3 E.8657 Each of the limits below represents an indeterminate form ; perform the appropriate algebraic trans-formations to determine each of its limits: a lim x ↦→ + ∞ x x − x c lim x ↦→ 2 − x − 2 2 x − 2 E.3435 Determine the value of the limits: a lim x ↦→−∞ x 2 − 7 + x b lim x ↦→ 1 − x 2 − 1 5 − x − 2 E.8660 Determine limit: lim x ↦→ + ∞ x x − x + 1 13. Limits and radicals: factoring E.6757 Consider the function f defined on R by: f ( x ) = 2 · x + 1 x 2 + 1 1 Justify that the function f is defined on R . 2 Establish the following identity: f ( x ) = x · 2 + 1 x | x |· 1 + 1 x 2 3 Deduce the following two limits: lim x ↦→−∞ f ( x ) ; lim x ↦→ + ∞ f ( x ) E.8658 Determine limit: lim x ↦→ 3 − 3 − x 2 x 2 − 5 x − 3 E.8659 Each of the limits below has an inde-terminate form. Demonstrate the expected result by showing the correct algebraic transformation : a lim x ↦→ 1 − 1 − x 1 − x = 0 b lim x ↦→ 1 − x − 1 x − 1 = 1 2 c lim x ↦→ - 2 3 + 2 x − 1 3 x + 2 · 3 x + 2 = - ∞ E.3454 1 Show that, for x ∈ −∞ ; − 1 √ 3 , we have : 3 x 2 − 1 + x = x · 1 − 3 − 1 x 2 2 Deduce the value of the following limit: lim x ↦→−∞ 3 x 2 − 1 + x E.5003 Consider the function f defined by the relation: f : x ↦−→ x 3 + x 2 x 1 Determine the definition set of the function f . 2 a For x ∈ − 1 ; 0 , establish equality: f ( x )= − x +1 b Determine the value of the limit lim x ↦→ 0 − f ( x ) . 3 Determine the value of the limit lim x ↦→ 0 + f ( x ) . 4 Can we talk about the limit of the function f in 0 . E.8661 Determine the value of the follow-ing limits: a lim x ↦→ + ∞ x 2 + x + 1 x − 1 b lim x ↦→−∞ x 2 + 1 x E.8663 Determine limit: lim x ↦→−∞ x 4 · x 2 − 3 x + 1 https://chingmath.fr chapExoCorrec/5026 sacados/5026 chapExoCorrec/3641 sacados/3641 chapExoCorrec/2960 sacados/2960 chapExoCorrec/4991 sacados/4991 chapExoCorrec/3338 sacados/3338 chapExoCorrec/8657 sacados/8657 chapExoCorrec/3435 sacados/3435 chapExoCorrec/8660 sacados/8660 Fait intervenir la valeur absolue pour simplification chapExoCorrec/6757 sacados/6757 chapExoCorrec/8658 sacados/8658 chapExoCorrec/8659 sacados/8659 chapExoCorrec/3454 sacados/3454 Fait intervenir la valeur absolue pour simplification chapExoCorrec/5003 sacados/5003 Fait intervenir la valeur absolue pour simplification chapExoCorrec/8661 sacados/8661 Fait intervenir la valeur absolue pour simplification chapExoCorrec/8663 sacados/8663
-101234567-1123ijJKABTC -6-5-4-3-2-123456I-1234JO 14. Asymmotes E.3434 Consider the function f defined by: f ( x ) = 4 x − 4 x + 1 1 Determine the definition set of the function f . 2 Study the limits of the function f at the bounds of its defining set. 3 Specify whether the curve C f , representative of the func-tion f admits asymptotes ; their characteristics will be specified. E.5714 By plotting the representative curves of functions using a calculator or plotting software, make a conjecture about the set of definition and the asymptotes to the curve of each of the functions below : a f ( x ) = 1 x 2 + 1 b g ( x ) = x + 1 x 2 − 1 c h ( x ) = x 2 − 3 x + 2 x − 1 d j ( x ) = ( x 2 +1) x 2 − 2 x +1 x − 1 15. Study of rational fractions E.3303 On the figure below, we have drawn the representative curve C of a function f derivable on − 3 2 ; + ∞ . Points J − 3 2 ; − 3 2 , K ( − 1 ; 0) , A 1 ; 11 4 , B (2 ; 2) are points of C ; The tangent at C in A is parallel to the x-axis. The tangent at C in B passes through T (4 ; 0) . The straight line of equation y =1 is asymptote to C in + ∞ . The function f is strictly increasing on − 3 2 ; 1 and strictly decreasing on 1 ; + ∞ . 1 Give the values of f − 3 2 , f ( − 1) , f (1) , f (2) as well as the limit of f in + ∞ . 2 Give, justifying your answers, the numbers f (1) and f (2) E.5004 Consider the function f defined by: f : x ↦−→ x 2 + 4 x − 1 x 2 − 2 x + 5 1 Determine the definition set of the function f . 2 a Determine the limits of the function f at the bounds of its defining set. b What asymptotes does the function f admit? 3 a Establish that the function f derived from the func-tion f admits the expression : f ( x ) = − 6 x 2 + 12 x + 18 x 2 − 2 x + 5 2 b Draw up the table of variations of the function f . 4 Determine the slope-intercept formof the tangent (Δ) to the curve C f at the point of abscissa 1 . 5 Draw the straight line (Δ) , the asymptotes to the curve C f and then the curve C f in the reference frame O ; I ; J orthonormal below : https://chingmath.fr chapExoCorrec/3434 sacados/3434 chapExoCorrec/5714 sacados/5714 chapExoCorrec/3303 sacados/3303 -101234567-1123ijJKABTC chapExoCorrec/5004 sacados/5004 -6-5-4-3-2-123456I-1234JO
-6-4-20246-224C1 -6-4-20246-6-4-2C2 -6-4-20246-224C3 -6-4-20246-224C4 -2-123456I-3-2-12345JO E.5025 Consider the function f defined on R by the relation: f ( x ) = x 3 − 4 x 2 + x + 12 4 · x 2 + 4 Note C f the representative curve of the function f in an or-thonormal frame. 1 Determine the limits of the function f in −∞ and in + ∞ . 2 a Determine the value of the reals a and b verifying the equality: f ( x ) = a · x + b + 16 4 · x 2 + 4 b Note ( d ) the straight line with equation : y = 1 4 · x − 1 . Study the relative position of the curve C f and the straight line ( d ) . 3 Determine the value of the limit: lim x ↦→ + ∞ f ( x ) − 1 4 · x − 1 4 Below are shown the four curves C 1 , C 2 , C 3 , C 4 . Which of these is the curve C f representative of the function f ? E.3433 Let f be defined on the set − 2 ; 1 ∪ 1 ; + ∞ and whose image of a number x is defined by the relation: f ( x ) = − ( x − 2) 2 2 · ( x − 1) · ( x + 2) In the plane with a reference frame O ; I ; J , note C f the representative curve of the function f . 1 a Determine the limits of the function f at the bounds of its defining set. b Specify any asymptotes to the curve C f and their char-acteristics. 2 Note ( d ) the straight line with equation y = − 1 2 . a Determine the value of the reals a , b , c realizing the following equality: f ( x ) = a + b · x + c 2 · ( x − 1) · ( x + 2) b Deduce the position of the curve C f relative to the straight line ( d ) on the interval 1 ; + ∞ 3 a Determine the expression of the derivative function f of the function f . b Justify that the curve C f admits horizontal tangents at the abscissa points 2 5 and 2 . c Draw up the table of variations of the function f ; it will be assumed that the image of 2 5 by the function f is 8 9 . 4 Determine the equation of the tangent ( T ) to the curve C f at the point of abscissa 0. 5 Draw the curve C f in the reference frame given in ap-pendix ; represent the asymptotes and horizontal tan-gents to the curve C f and the straight lines ( d ) and ( T ) . https://chingmath.fr chapExoCorrec/5025 sacados/5025 -6-4-20246-224C1 -6-4-20246-6-4-2C2 -6-4-20246-224C3 -6-4-20246-224C4 chapExoCorrec/3433 sacados/3433 fichierPlus/3433/diapoCorrection.pdf -2-123456I-3-2-12345JO
-4-3-2-1234I-4-3-2-1234JOCf(d -3-2-10123-4-3-2-11234ij E.6802 Consider the function f defined on R \ − 2 3 by: f ( x ) = 3 · x 2 + 5 · x + 4 3 · x + 2 We will note C f the representation of the function f in the reference frame O ; I ; J below : The straight line ( d ) is the representation of the function g defined by: g ( x ) = x + 1 1 a Give, without justification, the limits of the function f at the bounds of its defining set. b Specify whether the representative curve C f of the function f admits asymptotes. 2 a Determine the real a , b and c realizing the equality: f ( x ) = a · x + b + c 3 · x + 2 b Deduce that the function f admits as derivative the function f defined by: f ( x ) = 9 · x 2 + 12 · x − 2 3 · x + 2 2 c Draw up, justifying your approach, the table of varia-tions of the function f . We will only indicate the value of the extremums of f 3 For any natural number n , consider : M n the point of ( d ) with abscissa n , N n the point of C f of abscissa n , S n the segment [ M n N n ] . a Graph the segments S 0 , S 1 and S 2 . b Give the exact measure of segment S 0 . c What can be said about the length of segment [ M n N n ] when the value of n tends towards + ∞ . E.3639 Consider the function f defined by: f ( x ) = − 6 x 2 − 14 x + 360 ( x + 10)( x + 9) 1 Determine the values of x for which the image of x by the function f is strictly positive. 2 Determine the values of the following limits: lim x ↦→−∞ f ( x ) ; lim x ↦→ + ∞ f ( x ) 16. Studies of exponential functions E.3618 Consider the function f defined by the relation: f ( x ) = 1 e x − 1 We call C f the representative curve of the function f . 1 Determine the definition set of the function f . 2 Establish the table of variations of the function f . 3 Specify the various asymptotes of the curve C f . 4 Draw the curve C f . https://chingmath.fr chapExoCorrec/6802 sacados/6802 -4-3-2-1234I-4-3-2-1234JOCf(d chapExoCorrec/3639 sacados/3639 chapExoCorrec/3618 sacados/3618 -3-2-10123-4-3-2-11234ij
-123I-12JO E.5851 Let f be the function defined for any real x in the interval 0 ; 1 by: f ( x ) = 2 x − 2e − x + 1 e 1 a Draw up the table of variations of the function f on the interval 0 ; 1 . The exact values of f (0) and f (1) will be specified. b Show that the function f cancels once and only once on the interval 0 ; 1 in a real ¸ . Give the value of ¸ rounded to the hundredth. 2 Solve the following equation in the interval 0 ; 1 : x − e − x + 1 = e − x − x − e − 1 + 1 E.3677 Let f be the function defined on R by: f ( x ) = 9 2 · e − 2 x − 3 · e − 3 x We call C f the representative curve of f in an orthonormal reference frame O ; −→ i ; −→ j of unit 1 cm . 1 Show that for any x of R , we have : f ( x ) = 3 · e − 2 x · 3 2 − e − x . 2 Determine the limit of f in + ∞ then the limit of f in −∞ . 3 Study the variations of the function f and draw up the table of variations of f . 4 a Determine the coordinates of the intersection point of the curve C f with the y-axis. b Justify that the curve C f intercepts the x-axis at a sin-gle point. Give the approximate value of the coordinates of this point of intersection. 5 Calculate f (1) and plot the curve C f in the reference frame below : E.3654 Let f be the function defined on R by: f ( x )= 9 2 · e − 2 x − 3 · e − 3 x 1 Show that for any x of R , we have : f ( x ) = 3e − 2 x · 3 2 − e − x 2 Determine the limit of f in + ∞ then the limit of f in −∞ . 3 Study the variations of the function f and draw up the table of variations of f . E.4299 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider the func-tion f defined on R by: f ( x ) = x 2 · e − x We note f the derivative function of f 1 Determine the limits of the function f in −∞ and + ∞ . 2 Calculate f ( x ) and draw up the table of variations of f . 3 Deduce the sign of f at R . E.3658 Consider the function f defined on R by: f ( x ) = ( x + 1) · e − x Let ( C ) be its graphical representation in an orthonormal ref-erence frame O ; −→ i ; −→ j of the plane. The graphic unit is 4 cm . Study the variations of the function f and the limits of its defining set. Summarize these elements in a table of varia-tions as complete as possible. 17. Study of square root functions E.3341 Consider the function f defined on the interval − 1 ; + ∞ by the relation: f ( x ) = x 2 + 1 x + 1 Note C f the representative curve of the function f . 1 a Study the limits at the bounds of its defining set. b Does the curve C f admit asymptotes? if so, specify. 2 Note ( d ) the straight line with equation y =1 . Determine the relative position of C f and ( d ) . https://chingmath.fr chapExoCorrec/5851 sacados/5851 chapExoCorrec/3677 sacados/3677 Extrait Antilles Guyanne Juin 2008 -123I-12JO chapExoCorrec/3654 sacados/3654 Extrait Antilles Guyanes Juin 2008 chapExoCorrec/4299 sacados/4299 chapExoCorrec/3658 sacados/3658 Asie Juin 2008 chapExoCorrec/3341 sacados/3341
-1234567I-12345JO -4-3-2-1234I-3-2-12JOCgCf 3 a Establish the following equality: f (1+ h ) − f (1) h = 2 h 2 · h + 2 · 2 h 2 + 2 h + 2 + 2 · ( h + 2) b Deduce the value of the derived number f (1) . 4 Draw in the reference frame below the curve C f . 18. Composed of functions E.5008 Consider the two functions f and g de-fined on − 4 ; 4 by the relations : f ( x ) = 2 x + 1 ; g ( x ) = x 2 − 3 We give the curves C f and C g representative of the functions f and g in the reference frame O ; I ; J below : 1 By graphical reading, complete the following tables of values : x − 3 2 − 1 − 1 2 0 1 2 f ( x ) x − 2 − 1 0 1 2 g ( x ) 2 Consider the following calculation program : Take a number x ; Determine the image of x by the function f ; we note this number x ; Determine the image of x by the function g ; we note this number g f ( x ) . On peut noter ce programme de calcul par la chaine : x f , −−−−−→ f ( x ) g , −−−−−→ g f ( x ) a Determine the values of the following expressions : g f ( − 1) ; g f − 1 2 b Complete the following table of values : x − 3 2 − 1 − 1 2 0 1 2 g f ( x ) We’ve just created a new function which associates the image g f ( x ) with a number x . This function is called the function composed of f by g and is noted g ◦ f . 3 Draw in the reference frame below the curve C g ◦ f repre-sentative of the function g ◦ f . 4 Give the expression, as a function of x , of the function g ◦ f . https://chingmath.fr -1234567I-12345JO chapExoCorrec/5008 sacados/5008 -4-3-2-1234I-3-2-12JOCgCf
-3-2-123I-3-2-123JOCf -3-2-123I-3-2-123JOCg -4-3-2-1234I-4-3-2-123JOCf E.5009 Consider two functions f and g defined on the interval [ − 3 ; 3] whose representative curves, respec-tively C f and C g , are given in a reference frame O ; I ; J orthonormal : 1 Determine the value of the following expressions : a f ◦ g ( − 2) b f ◦ g (1.5) c f ◦ g (2) 2 Determine the value of the following expressions : a g ◦ f ( − 3) b g ◦ f (0) c g ◦ f (1) E.5010 Consider the function f defined on the interval − 4 ; 4 whose representative curve C f is given below in the ( O ; I ; J ) orthonormal coordinate system : 1 Calculate the following images : a f ◦ f (1) b f ◦ f ( − 2) c f ◦ f (3) 2 We define the function f n as the function composed n times of the function f by itself. Determine the value of the following images : a f 3 (1) b f 3 ( − 3) c f 4 ( − 1) E.3309 For each question, determine an expres-sion ˇ simplifiée ı of the expression of the compound f ◦ g of the function g by the function f : a f ( x ) = 2 x 2 − x + 1 ; g ( x ) = 3 x − 2 b f ( x ) = x − 2 ; g ( x ) = 4 x 2 + 12 x + 11 c f ( x ) = 1 x ; g ( x ) = 3 x + 1 2 − x d f ( x ) = x 2 − x + 1 ; g ( x ) = x e f ( x ) = x + 1 x − 1 ; g ( x ) = 1 x 19. Limits of function composites E.3355 For each question, determine the limit of g ◦ f in a : a f ( x ) = 2 x 2 − 5 x − 3 ; g ( x ) = 5 − x x 2 ; a = 3 b f ( x ) = 1 x + 3 ; g ( x ) = x +1 − x ; a = − 3 c f ( x ) = cos x − 2 x ; g ( x ) = x 3 + 2 x x 3 + x 2 ; a = + ∞ https://chingmath.fr chapExoCorrec/5009 sacados/5009 -3-2-123I-3-2-123JOCf -3-2-123I-3-2-123JOCg chapExoCorrec/5010 sacados/5010 -4-3-2-1234I-4-3-2-123JOCf chapExoCorrec/3309 sacados/3309 chapExoCorrec/3355 sacados/3355
−134∞0∞-∞0∞Variationdefx -∞−10∞-∞31∞4Variationdegx -12345678I-2-1234JOCvCu E.3436 Consider the numerical functions f and g whose table of variations is given below : 1 Specify the definition set for each of these two functions. 2 Justify that the function f admits a unique zero. 3 Determine and justify the value of the following limits: a lim x ↦→ 4 − g ◦ f ( x ) b lim x ↦→ + ∞ g ◦ f ( x ) c lim x ↦→ + ∞ g ◦ f ◦ g ( x ) d lim x ↦→− 1 f ◦ g ( x ) E.5023 Let f be a function defined on 0 ; + ∞ verifying: f (0) = 0 ; lim x ↦→ + ∞ f ( x ) = + ∞ Consider the function g defined on 0 ; + ∞ by the relation: g ( x ) = 1 x · f 1 x 1 Determine the limit of the function g in 0 . 2 Determine the limit of the function g into + ∞ . 3 Note C the representative curve of the function g . What consequences can be deduced from the previous two ques-tions for the curve C ? 20. Limits and comparisons E.3349 Consider the two numerical functions u and v defined on R whose representative curves C u and C v are given in the reference frame belowbelow : 1 In the above reference frame, draw the curve C f represen-tative of a function f defined on R verifying the following property: For any x ∈ R , we have : u ( x ) f ( x ) v ( x ) 2 Suppose the functions u and v admit the following lim-its : lim x ↦→ + ∞ u ( x ) = 1 ; lim x ↦→ + ∞ v ( x ) = 1 Make a conjecture when the limit of the function f in + ∞ . E.3351 Consider the function f defined by the relation: f ( x ) = 5 x 2 + 2 − cos x 1 Justify that the function f is defined on R . 2 a Establish the following frame : 5 x 2 + 3 f ( x ) 5 x 2 + 1 b Deduce the value of the following limit: lim x ↦→ + ∞ f ( x ) E.3354 Determine the following limits: a lim x ↦→ + ∞ x + sin( x ) x b lim x ↦→ + ∞ x · 2 + cos x 4 + sin x E.8662 Determine the value of the limit below : lim x ↦→ + ∞ 3 + cos x 1 + x 21. Exponential function and boundary limit E.3614 Determine the values of the following limits: a lim x ↦→ + ∞ e − x +1 b lim x ↦→−∞ e 2 x +1 c lim x ↦→ + ∞ e − x 2 +1 d lim x ↦→ + ∞ e 1 x e lim x ↦→ 0 − e 1 x f lim x ↦→ 0 − e − 1 x E.3707 Establish the value of the follow-ing limits: a lim x ↦→−∞ x 4 + 1 x − 1 = −∞ b lim x ↦→ 0 e x − 1 e 2 x − 1 = 1 2 https://chingmath.fr chapExoCorrec/3436 sacados/3436 −134∞0∞-∞0∞Variationdefx -∞−10∞-∞31∞4Variationdegx chapExoCorrec/5023 sacados/5023 chapExoCorrec/3349 sacados/3349 -12345678I-2-1234JOCvCu chapExoCorrec/3351 sacados/3351 chapExoCorrec/3354 sacados/3354 chapExoCorrec/8662 sacados/8662 chapExoCorrec/3614 sacados/3614 chapExoCorrec/3707 sacados/3707 Extrait concours ECE Mai 2002
tempst(en jours)020406080100120140160180200220hauteur(en mètres)0.20.40.60.811.21.41.61.82 E.5847 We are interested in the evolution of the height of a corn plant as a function of time. The graph below represents this evolution. Height is in meters and time in days. We decide to model this increase by a logistic function of the type : h ( t ) = a 1 + b · e − 0.04 · t où a and b are positive real constants, t is the time variable expressed in days and h ( t ) denotes the plant height, expressed in meters. We know that initially, for t =0 , the seedling measures 0.1 m and that its height tends towards a limiting height of 2 m . Determine the constants a and b so that the function h corre-sponds to the growth of the corn plant under study. E.3706 Let f be the function defined by the relation: f ( x ) = 2e 2 x − e x e 2 x − e x + 1 1 Justify that the function f admits R as its defining set. 2 Determine the value of the limits of f in −∞ and + ∞ . 3 Establish that the function f admits as expression : f ( x ) = − e x · e 2 x − 4e x + 1 e 2 x − e x + 1 2 E.3665 Consider the function f defined for any real number x by: f ( x ) = 4 · e x e x + 7 Note C the representative curve of the function f . 1 Verify that for any real x : f ( x )= 4 1+7e − x 2 a Demonstrate that the curve C admits two asymp-totes whose equations are to be specified. b Show that the function f is strictly increasing on R . c Show that for any real x : 0 <f ( x ) < 4 . 22. Exponential function and growth comparison E.3661 Determine the value of the following limits: a lim x ↦→ + ∞ e x 2 · e − x b lim x ↦→−∞ e x · x 2 − x + 1 c lim x ↦→ + ∞ e x + 1 e − x − 1 d lim x ↦→−∞ e − 3 x x 2 + 1 e lim x ↦→−∞ e − x + 3 x + 1 f lim x ↦→−∞ e − 2 x − e − x E.3615 Determine the values of the following limits: a lim x ↦→−∞ ( x + 1) · e x b lim x ↦→−∞ e x x c lim x ↦→ + ∞ x + 1 e x d lim x ↦→ + ∞ e 2 x − 3 · e x + 1 e lim x ↦→−∞ e 2 x − 3 · e x + 1 f lim x ↦→ + ∞ e x − 1 e x + 1 E.3710 Consider the function f defined on R \ 1 by: f ( x ) = 2 ( x − 1) 2 · e x +1 x − 1 The aim of this exercise is to determine the following two limits: lim x ↦→ 1 − f ( x ) ; lim x ↦→ 1 + f ( x ) 1 Soit X = 2 x − 1 . Prove equality: 2 ( x − 1) 2 · e x +1 x − 1 = e 2 · X 2 · e X 2 Deduce the value of the limits sought. 23. Limits by identification with derived numbers E.3662 Determine the value of the following limits: E.3777 1 Consider the sequence ( u n ) defined for n integer greater than or equal to 1 by: u n = 1 + e 1 n + e 2 n + · · · + e n − 1 n Determine the limit of the sequence ( u n ) . 2 Determine the following limits: a lim x ↦→ + ∞ x · e − x x 2 + 1 b lim x ↦→ + ∞ x 2 − 2( x − 1)e x − 1 3 Let f be the function defined on R by: f ( x ) = 1 4 · e − x 2 · x 2 + x Determine the limits in −∞ and + ∞ of the function f . 4 Let g be the function defined on R ∗ by: g ( x )= x e x − 1 Can we extend g on R by continuity. https://chingmath.fr chapExoCorrec/5847 sacados/5847 tempst(en jours)020406080100120140160180200220hauteur(en mètres)0.20.40.60.811.21.41.61.82 chapExoCorrec/3706 sacados/3706 chapExoCorrec/3665 sacados/3665 chapExoCorrec/3661 sacados/3661 chapExoCorrec/3615 sacados/3615 chapExoCorrec/3710 sacados/3710 chapExoCorrec/3662 sacados/3662 chapExoCorrec/3777 sacados/3777
-4-3-2-1234I2345JO -4-2246810121416182022I-224681012JO 24. Sequences and functions E.3519 We propose to show that the re-lations : u 0 = − 3 ; u n = u n − 1 − 8 2 · u n − 1 − 9 pour tout n ∈ N ∗ define a sequence and that this sequence is convergent. 1 Consider the function f defined on −∞ ; 9 2 by: f ( x ) = x − 8 2 x − 9 Note C f the representative curve of the function f . a Study the limits of the function f at the bounds of its defining set. Name the asymptotes of the curve C f . b Determine the expression of the function f derivative of the function f . Deduce the table of variations of the function f . c Determine the equation of the tangent to the curve C at the point of abscissa 1. d Draw the curve C f in the reference frame below, and the straight line with equation y = x : e Using this graph, make a conjecture about the behav-ior of the sequence u n . 2 Show that for any n of N : u n < 1 . 3 Show that the sequence is increasing and converges. (The value of the limit is not asked) E.3416 We want to model the evolution of the number, expressed in millions, of French households owning a flat-screen TV, as a function of the year. Let u n be the number, expressed in millions, of households owning a flat-screen TV in year n . We posit n =0 in 2005 , u 0 =1 and, for any n 0 : u n +1 = 1 10 · u n · 20 − u n 1 Let f be the function defined on 0 ; 20 by: f ( x ) = 1 10 · x · 20 − x a Deduce that for any x ∈ 0 ; 20 : f ( x ) ∈ 0 ; 10 . b The representative curve C of the function f in an orthonormal frame is given in the appendix. Using this graph, represent on the x-axis the first five terms of the sequence u n n 0 . 2 Show by recurrence that for any n ∈ N : 0 u n u n +1 10 . 3 Show that the sequence u n n 0 is convergent. (the value of its limit is not asked) . E.3422 Consider the sequence v n de-fined on N by: v 0 = 6 v n +1 = 1.4 · v n − 0.05 · v n 2 pour tout n ∈ N . 1 Let f be the function defined on R by: f ( x ) = 1.4 x − 0.05 · x 2 a Study the variations of the function f on the interval 0 ; 8 . b Show by recurrence that, for any natural number n : 0 v n v n +1 8 2 Establish the convergence of the sequence v n (the value of the limit will not be asked) . E.5053 Consider the function f defined on R by: f ( x ) = 1 8 · x 2 + 1 2 · x + 1 2 1 Show that if x ∈ 0 ; 2 then f ( x ) ∈ 0 ; 2 2 In this question, any trace of research, however incom-plete, or initiative, however unsuccessful, will be taken into account in the assessment. We define the sequence u n by: u 0 = 1 2 ; u n +1 = f u n for any n ∈ N . Establish the convergence of the sequence u n (the value of the limit is not asked) . 25. Courses https://chingmath.fr chapExoCorrec/3519 sacados/3519 Paris 1985 -4-3-2-1234I2345JO chapExoCorrec/3416 sacados/3416 -4-2246810121416182022I-224681012JO chapExoCorrec/3422 sacados/3422 chapExoCorrec/5053 sacados/5053
E.5500 Establish the following two limits: lim x ↦→ + ∞ e x = + ∞ ; lim n ↦→−∞ e x = 0 E.3657 The result is assumed to be known Proposition: lim x ↦→ + ∞ e x x = + ∞ Prove that : lim x ↦→ + ∞ x · e − x =0 E.3282 Consider a function g continuous, strictly increasing on 0 ; + ∞ and such that lim x ↦→ 0 g ( x )= −∞ and lim x ↦→ + ∞ g ( x )=+ ∞ . We admit that we can define on N a sequence ( ˛ n ) of real numbers such that g ( ˛ n )= n , and that this sequence is strictly increasing. Show that the sequence ( ˛ n ) tends to + ∞ . 26. Share E.2526 Consider the function f defined by: f : x ↦−→ 5 x + 1 − 5 x 2 + 4 x + 1 1 Give the definition set of the function f . 2 Determine the limits of the function f in + ∞ and −∞ . 3 a Justify that the two limits below represent an inde- terminate form : lim x ↦→− 1 5 − f ( x ) and lim x ↦→− 1 5 + f ( x ) b Show that, for x ∈D f , we have : f ( x )= − 1 x − 1 c Deduce the value of the two limits presented in ques-tion a . 27. Unclassified financial years E.3368 Solve the following system of equations : a + 2 b − 4 c = 4 2 a − b + c = − 8 − 3 a − 2 b + c = − 3 E.10422 Consider the function f defined on − 3 ; + ∞ by: f : x ↦−→ x 2 + 2 x + 1 x + 3 1 Show that the number derivative of f in x is written : f ( x ) = 3 · x 2 + 14 · x + 11 2( x + 3) x + 3 2 Draw up the sign table for the function f . 3 Deduce the table of variations of the function f . 4 Give the minimum of the function f on its defining set. https://chingmath.fr chapExoCorrec/5500 sacados/5500 chapExoCorrec/3657 sacados/3657 Asie Juin 2008 chapExoCorrec/3282 sacados/3282 sacados/2526 chapExoCorrec/3368 sacados/3368 chapExoCorrec/10422 sacados/10422