Outside the high school program / Asymptotic behavior 41 exercises (including 25 corrected)

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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
xVariationdef−∞2∞3−∞∞−∞ xVariationdef−∞−23∞−∞−1−1∞ xVariationdef−∞14∞0−∞2∞−∞∞ 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∞−∞∞
-4-3-2-1234I-2-123JOCf -3-2-123456I-2-1234JOA 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
-4-3-2-12345I-3-2-123JOCf ijCf(dNMH 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
-3-2-123456I-2-1234567JO 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
x-4-20246810y-6-4-22468 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
-4-3-2-12I-2-1234JO 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.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.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/2524 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 ) https://chingmath.fr chapExoCorrec/2290 sacados/2290