Outside the high school program / Asymptotic behavior 41 exercises corrected

a
1. First approach E.2498 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 3 x − 1 2 x + 1 1 Give the definition set of this function. 2 By plotting the representative curve of this function on your calculator, make a conjecture about the value of the following limits: lim x ↦→ + ∞ f ( x ) ; lim x ↦→−∞ f ( x ) lim x ↦→− 1 2 + f ( x ) ; lim x ↦→− 1 2 − f ( x ) 3 a Establish the following algebraic relationship : f ( x ) = − 5 4 x + 2 + 3 2 b Deduct the value of the following limits: lim x ↦→ + ∞ f ( x ) ; lim x ↦→−∞ f ( x ) 4 a What can we say about the sign of 3 x − 1 when x approaches − 1 2 ? b Deduce the value of the following limits: lim x ↦→− 1 2 + f ( x ) ; lim x ↦→− 1 2 − f ( x ) E.2499 Consider the function f defined by: f ( x ) = 3 x 2 − x + 3 3 x − 1 1 Give the definition set of the function f . 2 a Draw the representative curve of this function on your calculator. b Using ˇ Zoom out ı, observe the shape of the curve in the vicinity of −∞ and + ∞ . What can you say about the shape of the curve? 3 a Establish the following algebraic relationship : f ( x ) = 3 3 x − 1 + x b Give the value of the following limits: lim x ↦→−∞ f ( x ) − x ; lim x ↦→ + ∞ f ( x ) − x 4 Deduce the observations made with the calculator. E.6175 Consider three functions f , g , and h defined on R ∗ + by the relations : f ( x ) = x + 2 x 2 + 3 ; g ( x ) = 2 · x − 1 √ x ; h ( x ) = 2 · x 2 + x 5 · x Below is a table of values for each of the functions : x 1 0.1 0.01 0.001 0.0001 f ( x ) 3 4 2.1 3.01 2.01 3.000 1 2.00 1 3.000 001 2.000 1 3.000 000 01 x 0.01 0.000 1 0.000 001 g ( x ) − 0.98 0.1 − 0.999 8 0.01 − 0.999 998 0.00 1 x 1 0.1 0.01 0.001 0.000 1 h ( x ) 3 5 0.12 0.5 0.010 2 0.05 0.001 002 0.005 0.000 100 02 0.000 5 Note that, in each table, the values of x ˇ progress slowly ı towards 0 . 1 a For each table, use a calculator to observe the pro-gression of the rounded decimal values of these quo-tients. b In each case, make a conjecture about the limit value of these images when : ˇ x tends towards 0 by values greater than 0 ı For the function f , this value is denoted by: lim x ↦→ 0 x> 0 f ( x ) or lim x ↦→ 0 + f ( x ) . 2 Using your calculator, plot the representative curves of these functions and observe the curve in the ˇ vicinity ı of the y-axis. 2. Handling indeterminate forms E.2514 1 a Explain why the following limit is an indeterminate form : lim x ↦→ + ∞ x 2 − x + 1 b Justifying the equality below, give the value of the limit from the previous question : x 2 − x + 1 = x 2 1 − 1 x + 1 x 2 2 a Explain why the following limit is an indeterminate form : lim x ↦→ + ∞ 2 x 2 + 2 x + 1 3 x 2 − 2 b Justifying the equality below, give the value of the limit from the previous question : 2 x 2 + 2 x + 1 3 x 2 − 2 = x 2 2 + 2 x + 1 x 2 x 2 3 − 2 x 2 3 The limit below is an indeterminate form ; using a suit-able transformation, determine the value of this limit: lim x ↦→ + ∞ − 2 x 2 + 1 2 x + 1 https://chingmath.fr sacados/2498 sacados/2499 chapExoCorrec/6175 sacados/6175 sacados/2514
E.2525 Consider the function f defined by: f : x ↦−→ x 2 + 5 x − 6 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 − f ( x ) and lim x ↦→ 1 + f ( x ) b Show that, for x ∈D f , we have : f ( x )= x +6 c Deduce the value of the two limits presented in ques-tion a . 3. Infinite limit calculations E.3055 Determine the limits below : a lim x ↦→−∞ x · (2 x + 1) b lim x ↦→ + ∞ (3 − 4 x )( x + 1) c lim x ↦→−∞ 3 x 2 + 2 x + 3 2 x + 1 d lim x ↦→ + ∞ 3 x − 4 5 + 1 x e lim x ↦→−∞ 3 2 x − 1 f lim x ↦→−∞ x 2 + x x 3 − 2 x E.3047 Among the limits proposed below, give those representing an indeterminate form ; give the value of the other limits: a lim x ↦→ + ∞ 2 x 4 − x 3 b lim x ↦→−∞ 5 x 3 − 2 x 2 c lim x ↦→ + ∞ 3 5 − 2 x d lim x ↦→−∞ 2 + x 2 e lim x ↦→ + ∞ x + √ x x 2 f lim x ↦→−∞ x 2 + 1 − x 4 + x E.3056 Determine the limits below : a lim x ↦→ + ∞ 3 x + 2 5 − 3 x b lim x ↦→ + ∞ x 2 + 3 x − 1 2 x + 1 c lim x ↦→−∞ 3 − 2 x x 2 − 4 x + 7 d lim x ↦→ + ∞ 5 x 2 − 3 5 − 3 x 2 e lim x ↦→−∞ x 4 + 3 x + 1 x 3 − 2 x 2 + 4 f lim x ↦→ + ∞ x 100 − 5 x 44 + x 14 3 x 102 − 5 x 56 E.3096 Determine each of the following limits: a lim x ↦→ + ∞ − 5 x 3 − 7 x b lim x ↦→ + ∞ 2 x 3 − 3 x 2 c lim x ↦→−∞ x 7 − 5 x 6 2 d lim x ↦→ + ∞ x 5 − x 3 + 2 − 2 x 2 − x + 2 e lim x ↦→−∞ x 3 − x 2 + 5 x 4 + 3 x + 1 f lim x ↦→−∞ x 2 − 5 x + 2 − 3 x 2 + 5 x − 1 4. Calculating limits in a real E.3057 Determine the limits below : a lim x ↦→ 1 2 − 3 2 x − 1 b lim x ↦→ 3 − − 2 3 − x c lim x ↦→ 3 2 + 4 − x 3 − 2 x d lim x ↦→ 5 − ( x − 5)(7 − x ) 2 x − 10 e lim x ↦→ 4 − 1 8 − 2 x f lim x ↦→− 2 − 3 x 2 + 4 x − 4 2 x + 4 E.3066 Determine, if possible, the limits below : a lim x ↦→ 0 + 1 x − 1 x 2 b lim x ↦→ 0 − 3 x 4 − 2 x 2 5 x 2 − x c lim x ↦→ 0 − 6 x 12 + 5 x 6 − 3 x 2 3 x 8 − 5 x 2 d lim x ↦→ 0 + 1 x sin x E.3048 Among the limits proposed below, deter-mine those representing an indeterminate form ; give the value of the other limits: a lim x ↦→ 0 + x 2 + 2 x b lim x ↦→ 2 x 2 − 4 x + 4 2 x 2 + x − 6 c lim x ↦→ 3 2 − 3 − 2 x d lim x ↦→ 0 x 3 − 2 x 2 x 2 − 3 x + 1 e lim x ↦→ 0 + 5 x 2 + x f lim x ↦→ 4 − 5 x + 3 − 2 x − 4 E.3102 Determine the following limits: a lim x ↦→ 3 + 1 ( x − 3)( x − 5) b lim x ↦→− 2 + 3 x − 2 − 2 x + 1 2 + x c lim x ↦→ 0 − 2 x 2 + x 1 x − 3 x 3 d lim x ↦→ 2 5 + 10 x 2 + x − 2 − 2 x 2 − 5 x − 2 E.3097 Determine the values of the following limits: a lim x ↦→ 0 x 4 − x 3 x 2 b lim x ↦→ 2 + x + 5 4 − 2 x c lim x ↦→− 1 x + 1 x 2 − 1 d lim x ↦→ 3 + x − 5 3 x 2 − 11 x + 6 e lim x ↦→ 1 3 + 3 x 2 + 2 x − 1 x − 1 f lim x ↦→ 2 + − 2 x 2 + 7 x − 6 x 2 − 4 x + 4 https://chingmath.fr sacados/2525 chapExoCorrec/3055 sacados/3055 sacados/3047 chapExoCorrec/3056 sacados/3056 chapExoCorrec/3096 sacados/3096 chapExoCorrec/3057 sacados/3057 chapExoCorrec/3066 sacados/3066 sacados/3048 chapExoCorrec/3102 sacados/3102 chapExoCorrec/3097 sacados/3097
5. Limit calculations E.2546 Determine the value of the following limits: a lim x ↦→ + ∞ x | x | b lim x ↦→−∞ | x | x c lim x ↦→ + ∞ x 2 + 1 x d lim n ↦→ + ∞ 2 x + 1 − 2 x e lim x ↦→ + ∞ 2 x + 1 − x f lim x ↦→− 1 + 1 x + 1 − 1 x 2 + 3 x + 2 E.2618 Determine the value of the following limits: a lim x ↦→ + ∞ 4 x 2 + 2 x − 1 − 2 x + 1 b lim x ↦→− 2 3 + 3 x 2 − 1 12 x 2 + 23 x + 10 c lim x ↦→− 2 − − x 2 + x + 6 4 x 2 + 16 x + 16 d lim x ↦→ 1 − 1 1 − x 2 − 1 (1 − x ) 2 e lim x ↦→ 4 + 2 − x x − 4 f lim x ↦→ 4 + 8 − x x x − 4 Hint for question f , determine the values of a , b , c verifying the following factorization : 64 − x 3 = ( x − 4)( a · x 2 + b · x + c ) E.2537 Determine the following limits: a lim x ↦→ + ∞ x + cos x b lim x ↦→ + ∞ 2 + cos x x c lim x ↦→ + ∞ x + sin x x + cos x E.2584 Determine the value of the following limits: a lim x ↦→−∞ 2 x 2 + 3 x − 1 3 x 3 − 2 b lim x ↦→ 1 2 + 3 x − 2 − 2 x 2 + 7 x − 3 c lim x ↦→− 2 − 2 x − x + 3 2 + x d lim x ↦→ 3 4 x 2 − 12 x + 9 − x 2 + 5 x − 6 e lim x ↦→−∞ x − 2 x 2 + 1 f lim x ↦→ + ∞ x − 2 x 2 + 1 E.3067 Determine, if possible, the limits below : a lim x ↦→ + ∞ x 2 − 3 x − x b lim x ↦→ 1 + 1 x − x 2 x − 1 c lim x ↦→ 0 + 1 x sin x d lim x ↦→ + ∞ x 2 + 1 − x e lim x ↦→− 3 − 6 − x − 3 2 x 2 + 5 x − 3 f lim x ↦→ 0 + 1 x − x x E.3098 Determine the value of the following limits: a lim x ↦→ + ∞ x 5 − 2 x 2 x 2 + x b lim x ↦→ + ∞ x 2 − 4 − x c lim x ↦→ 4 − 4 − 2 x x − 4 d lim x ↦→ + ∞ x 3 + 2 − x e lim x ↦→ 1 − 1 − x 2 x 2 − 4 x + 2 E.3101 Determine the value of the following limits: a lim x ↦→−∞ − 3 x 4 + 2 x b lim x ↦→ + ∞ 6 x 2 − 1 x 3 x 2 − 2 x c lim x ↦→ + ∞ x 6 − 3 x 2 3 x 2 + 7 x d lim x ↦→ + ∞ 3 x − x 2 x + 3 E.2513 If possible, give the value of the following limits; indicate the indeterminate forms among them : a lim x ↦→ + ∞ 1 x − 2 b lim x ↦→ 2 2 x + 1 3 x 2 − 3 x − 2 c lim x ↦→− 5 x 2 ( x + 5) 2 d lim x ↦→− 2 − 1 − x − 2 e lim x ↦→ + ∞ x 2 − x + 1 f lim x ↦→− 1 + x − 3 2 x + 2 6. Horizontal and vertical asymptote E.2534 Looking at each of the tables of varia-tions below, write the corresponding limits at the boundaries of its set of definitions and specify whether the function has horizontal or vertical asymptotes : a b c https://chingmath.fr chapExoCorrec/2546 sacados/2546 chapExoCorrec/2618 sacados/2618 sacados/2537 chapExoCorrec/2584 sacados/2584 chapExoCorrec/3067 sacados/3067 chapExoCorrec/3098 sacados/3098 chapExoCorrec/3101 sacados/3101 sacados/2513 sacados/2534 xVariationdef−∞2∞3−∞∞−∞ xVariationdef−∞−23∞−∞−1−1∞ xVariationdef−∞14∞0−∞2∞−∞∞
E.3074 Below is the representative curve C f of the function f defined on the interval − 4 ; 4 : The horizontal and vertical asymptotes to the C f curve have been shown in dotted lines. Draw up the complete table of variations of this function. 7. Oblique asymptote E.2536 1 a Study the following limit: lim x ↦→ + ∞ x 2 + 3 x − 2 2( x + 1) − 1 2 x − 1 b Give the equation of the oblique asymptote of the func-tion f in + ∞ defined by: f ( x ) = x 2 + 3 x − 2 2( x + 1) 2 a Determine the values of a and b so that : a · x + b + 2 x + 2 = − 6 x 2 − 11 x + 8 3( x + 2) b Determine the equation of the oblique asymptote in + ∞ of the function g defined by: g ( x ) = − 6 x 2 − 11 x + 8 3( x + 2) E.3099 1 Consider the function f defined whose image of a number x is defined by: f ( x ) = 6 x 3 + x 2 + 7 x + 9 2 x 2 − x + 3 Show that the straight line ( d ) of equation y =3 x +2 is the asymptote to the curve C f at −∞ and + ∞ . 2 Consider the function g defined by the relation: g ( x ) = 4 x 2 − 8 x + 5 3 − 2 x Determine the values of the real numbers a and b verify-ing : lim x ↦→ + ∞ g ( x ) − ( a · x + b ) = 0 E.3103 Consider the function f whose image of a number x is defined by: f ( x ) = 8 x 2 − 2 x − 5 2 x + 1 1 Determine the value of the reals a , b , c verifying the equality: f ( x ) = a · x + b + c 2 x + 1 for all x ∈ R \ − 1 2 2 Justify that the curve C f of the function f admits an oblique asymptote whose equation is to be specified. E.3356 In the plane marked with a reference point O ; I ; J , the representative curve C f of a function f defined on − 3 ; 6 \ 1 The asymptotes to the curve C f are shown as dotted lines. 1 Specify the nature and equation of each of the asymp-totes to the curve C f . 2 Give the value of each of the following limits: lim x ↦→−∞ f ( x ) ; lim x ↦→ 1 + f ( x ) ; lim x ↦→ + ∞ f ( x ) https://chingmath.fr chapExoCorrec/3074 sacados/3074 -4-3-2-1234I-2-123JOCf sacados/2536 chapExoCorrec/3099 sacados/3099 chapExoCorrec/3103 sacados/3103 chapExoCorrec/3356 sacados/3356 -3-2-123456I-2-1234JOA
E.3350 Consider the function f defined on R whose image of a real number x is defined by the relation: f ( x ) = x 3 − 2 x 2 + x + 4 2( x 2 + 1) Below is the curve C f representing the function f in the co-ordinate system O ; I ; J : 1 a Determine three real numbers a , b and c that satisfy the relation: f ( x ) = a · x + b + c x 2 + 1 b Deduce the equation of the oblique asymptote (Δ) to the curve C f at −∞ and + ∞ . c Draw the line (Δ) 2 Consider a function g defined on R whose representative curve C g has relative position with C f : C g is below C f on −∞ ; 1 ; C g is above C f on 1 ; + ∞ . a Plot a function g satisfying the above conditions. b Make a conjecture about the value of the following two limits: lim x ↦→−∞ g ( x ) ; lim x ↦→ + ∞ g ( x ) 8. Oblique asymptote and study E.2619 Consider the function f defined on R and whose image of a real number x is given by the relation: f ( x ) = 2 x 2 (2 x + 3) 2 x 2 + 2 x + 1 1 a Determine the two real numbers a and b verifying the relation: f ( x ) = a · x + b − 4 x + 1 2 x 2 + 2 x + 1 b Show that the curve C f of the function f admits an oblique asymptote ( d ) in + ∞ and in −∞ whose re-duced equation will be specified. 2 Here is a representation of this curve and its oblique asymptote : Consider the point M on the curve C f with abscissa x and N is the point on the line ( d ) with abscissa x . The point H is the orthogonal projection of these two points on the ordinate axis; its coordinate is H ( x ; 0) . In the next questions, we study the distance MN for values of x belonging to R + : a Express the distance MN as a function of x . b Show that for any x ∈ R + , we have : ⏐ ⏐ ⏐ ⏐ 4 x + 1 2 x 2 + 2 x + 1 ⏐ ⏐ ⏐ ⏐ 2 x Hint : we can use the fact that each term is positive on R + c Deduce the existence of an interval of the form [ a ; + ∞ [ on which MN 10 − 6 . E.2585 Consider the function f whose image of x is defined by the relation: f ( x ) = 4 x 2 + 10 x − 2 x + 3 1 Determine the limits of this function at the boundaries of its defining set. 2 a Determine the value of the three real numbers a , b , c verifying: a · x + b + c x + 3 = 4 x 2 + 10 x − 2 x + 3 b Deduce an expression for the function f derived from the function f . c Draw up the complete table of variations of the func-tion f (do not omit any values from the table) . 3 Determine the equation of the oblique asymptote in + ∞ . https://chingmath.fr chapExoCorrec/3350 sacados/3350 -4-3-2-12345I-3-2-123JOCf chapExoCorrec/2619 sacados/2619 ijCf(dNMH chapExoCorrec/2585 sacados/2585
E.2535 Consider the function f whose image of a number x is given by the relation: f ( x ) = 6 x 2 + x + 1 2 x + 1 1 Determine the definition set of the function f . a Determine the expression of the derivative function of f . b Draw up the table of signs of the function f . c Deduce the table of variations of the function f . d By studying the limits of the function f at the bounds of its defining set, complete the table of variations. 2 Show that the function f admits in −∞ and in + ∞ the straight line ( d ) of equation y =3 x − 1 as its oblique asymptote. E.3306 Let f be a function defined and derivable on the interval − 3 ; + ∞ , increasing on the intervals − 3 ; − 1 and 2 ; + ∞ et decreasing on the interval − 1 ; 2 . We denote f its derivative function on the interval − 3 ; + ∞ . The curve Γ representative of the function f is plotted below in an orthogonal frame of reference O ; −→ i ; −→ j . It passes through the point A ( − 3 ; 0) and admits as its asymp-tote the straight line Δ of equation y =2 x − 5 The answers will not be justified. Scoring : a correct answer earns 0.5 points ; an incorrect an-swer deducts 0.25 points no answer earns no points and deducts no points. If the total number of points is nega-tive, the overall score for the exercise is 0 . 1 The equation f ( x )=4 has exactly two solutions in the interval − 3 ; + ∞ . 2 lim x ↦→ + ∞ f ( x ) = + ∞ 3 lim x ↦→ + ∞ ⏐ ⏐ ⏐ f ( x ) − (2 x − 5) ⏐ ⏐ ⏐ = + ∞ . 4 f (0) = 1 5 f ( x ) > 0 for any real number x belonging to the interval − 2 ; 1 E.3581 Let f be the function defined on 0 ; + ∞ par : f ( x ) = x 3 + 3 2 · x and let C la courbe représentative de f dans un repère or-thonormé O ; I ; J . 1 a Étudier variations in f sur the interval 0 ; + ∞ . b Precise the equations of the asymptotes of C (to determine one of these asymptotes, study lim x ↦→ + ∞ f ( x ) − x 3 ) . c Trace the curve C . 2 a So m a real number and let Δ la be the line with equation y = m . Discuss, depending on the values of m , the number of points of intersection of Δ and C . b For any m> 2 , we call A et B les points of intersec-tion of Δ et of C . Let I be the middle of segment [ AB ] . Show that, when m describes the interval √ 2 ; + ∞ , I décrit a part, to be specified, of the straight line D d equation x = 3 2 · y . 9. Framing E.2545 Consider the function f defined by: f : x ↦−→ − 2 x 2 − 2 x − 3 x 2 − 2 x − 3 1 Determine the defining set D f . 2 Determine the limits of the function f in −∞ and + ∞ . 3 a Draw up the sign table for the expression x 2 − 2 x − 3 . b Deduce the value of the following limits: lim x ↦→− 1 − f ( x ) ; lim x ↦→− 1 + f ( x ) lim x ↦→ 3 − f ( x ) ; lim x ↦→ 3 + f ( x ) 4 a Establish that the derivative of the function f is : f ( x ) = 6 x ( x + 3) ( x − 3)( x + 1) 2 https://chingmath.fr sacados/2535 chapExoCorrec/3306 sacados/3306 -3-2-123456I-2-1234567JO chapExoCorrec/3581 sacados/3581 Aix-Marseille 1988 sacados/2545
b Draw up the complete table of variations of the func-tion f . 5 a Solve the inequation : f ( x ) 100 b Does the result of the previous question match those of question 3 ? Justify your answer. 6 a Solve the inequation : f ( x )+2 6 × 10 2 b Solve the inequation : − 6 × 10 − 2 f ( x )+2 c Deduce the solutions of the following frame : | f ( x ) + 2 | 6 × 10 − 2 7 Do the results found in the previous question agree with the results found in question 2 . 10. Limits and homographic functions E.593 1 In the reference frame below, draw the representative curve of the function f : x ↦−→ 2+ x x − 3 . Here’s one of Zeno of Elea’s paradoxes (500 - 430 BC) : ˇThere is no movement, as the mobile must reach the middle of its path before reaching the endı 2 a So, we are going to move on a graduated line from the point A (4) to the point B (3) : we’ll say it’s a move to the left, but also that we’re moving towards 3, but staying with values greater than 3, we’ll note x ↦−→ 3 + . Moving in the manner of Zeno of Elea, we note : u 0 the initial position abscissa : i.e. 4; u 1 the abscissa of the remaining halfway point : 3.5; u 2 the abscissa of half the remaining path : 3.25; . . . Complete the following table : n 0 1 2 3 4 5 6 7 u n 4 3.5 3.25 b Verify, using your calculator’s functions and value ta-ble, that the value of u n can be expressed as a func-tion of the value of n by the following relationship (functional) : u n = 3 + 1 2 n c Give, for the first three precisions requested in the ta-ble below, from which value of n , u n is an approximate value of 3: Précision 10 − 1 10 − 2 10 − 3 10 − 10 Value of n d To use this formula in the table, we’ll transform it: 3 + 1 2 n − 3 < 0.5 × 10 − 10 3 + 1 2 n − 3 × 10 10 < 0.5 1 2 n × 10 10 < 0.5 Determine the smallest natural integer n verifying this inequality. e Is there a n verifying u n =3 ? Can we say that there exists a rank N from which u n becomes an approximate value of 3 to the nearest 10 − 100 . Note that lim n ↦→ + ∞ u n =3 . This means that the u n posi-tion will be as close to 3 as desired, as long as the n value is increased. 3 a Show that : f ( x )=1+ 5 x − 3 b Let n be an integer, posing x =3+ 1 2 n . Give the expres-sion of f ( x ) as a function of n . Simplify this entry. c Complete the following table with the exact values : x 4 3.5 3.25 3.125 3.0625 3.03125 f ( x ) d What can we say about the value of f ( x ) as x ap-proaches more and more towards 3 from the left. We’ll note this value lim x ↦→ 3 + f ( x ) . e What can be said about the relative position of the rep-resentative curve of f and the straight line of equation x =3 . We will say that the straight line with equa-tion x =3 is a vertical asymptote to the representative curve C f 4 What can we say about lim x ↦→ 3 − f ( x ) ; i.e. the limit value of the image of x when x towards 3 by the right (keeping values less than 3) . We will now study the behavior of the function f when x goes to + ∞ . 5 a Consider the number sequence defined by v n =3+2 n . Complete the table below : https://chingmath.fr sacados/593 x-4-20246810y-6-4-22468
n 1 2 3 4 5 v n b Complete the following table with the correct values : x v 1 v 2 v 3 v 4 v 5 f ( x ) c What can be the value of f when x tends towards + ∞ . We’ll note this value lim x ↦→ + ∞ f ( x ) (the limit value of the image of x by the function f when x tends to + ∞ ) d What can be said about the relative position of the curve with respect to the line of equation y =1 ? We’ll say that the straight line with equation y =1 is a hori-zontal asymptote to the representative curve C f . e Quickly imagine what the value of lim x ↦→−∞ f ( x ) will be. E.594 A homographic function is any function whose algebraic expression is of the form a · x + b c · x + d , where a , b , c , and d are fixed real numbers. 1 a For what value of x is this function undefined? b What can you say in the case where c =0 and d =0 ? We accept the following proposition : Proposition: Any homographic function x ↦−→ a · x + b c · x + d can be written in the form x ↦−→ ¸ + ˛ c · x + d 2 For each of the functions below, determine its domain, as well as their values of ¸ and ˛ : g : x ↦−→ 3 x + 2 x + 1 ; h : ↦−→ x 1 − 2 x k : x ↦−→ 2 x − 4 5 + 2 x 3 Deduce the equation of their horizontal and vertical asymptotes for each of them. 11. Unclassified financial years E.2500 y = 1 = ( − x + 1) + 2 = (( x + 2) 2 + 1) E.2524 For each of the functions below, determine their definition set and then at the bounds of their definition set, determine the limits ˇ to the left ı and ˇ to droite ı. f : x ↦−→ 3 x + 1 x − 2 ; g : x ↦−→ 1 (2 x − 1)(3 x + 5) h : x ↦−→ x + 2 4 x 2 + 4 x + 1 ; j : x ↦−→ 2 x − 4 x 2 − 1 k : x ↦−→ x + 2 2 x 2 + 5 x + 2 E.2586 Doing something with 2 x + 2 + 2 x − 4 e whose derivative is 2( x − 4)( x − 2) ( x − 3) 2 Great for studying derivatives and sign tables. E.2589 After drawing up the variation ta-ble, give the maximum draw the curve to make exo y = (4 x 2 − 12 x + 9) = ( − x 2 + 5 x − 6) E.3106 exercise the limit form in 0 2 x 2 + x 1 x − 3 x 3 and 2 x 2 + x 1 x − 3 x 3 E.3652 Let f be the function defined on R by: f ( x )=e x − x − 1 and let ( C ) be its representative curve in an orthonormal plane frame. The line ( D ) of equation y = − x − 1 is asymptotic to ( C ) . Shown below is the curve ( C ) and the straight line ( D ) . 1 Let a be a real number. Write, as a function of a , an equation of the tangent ( T ) at ( C ) at the point M of abscissa a . 2 This tangent ( T ) intersects the line ( D ) at the point N of abscissa b . Check that : b − a = − 1 . 3 Deduce a construction, to be carried out on the attached sheet, of the tangent ( T ) to ( C ) at the point of abscissa https://chingmath.fr sacados/594 sacados/2500 sacados/2524 sacados/2586 sacados/2589 sacados/3106 chapExoCorrec/3652 sacados/3652 Extrait Nouvelle-Caledonie Novembre 2007 -4-3-2-12I-2-1234JO
1.5 . We will show the corresponding N point. E.2290 Consider the function f defined on R ∗ by the relation: f ( x ) = 3 · x 3 + 2 · x 2 · x 3 − 2 · x 2 + x 1 Let g be the function defined on R by: g ( x ) = 3 · x 2 + 2 2 · x 2 − 2 · x + 1 Show that f is the restriction on R ∗ of the function g . 2 Deduce the limit of the function f in 0. That is, the value of lim x ↦→ 0 x =0 f ( x ) E.3576 Study the function f defined by: f ( x ) = x 2 + 2 x + 3 − x + 4 x 2 + 1 It has an oblique asymptote, a horizontal asymptote and a super horizontal tangent for the trace E.3577 study of √ x 2 x + 1 ?? https://chingmath.fr chapExoCorrec/2290 sacados/2290 sacados/3576 sacados/3577