Section 4.5 Enhanced Assignment

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 1. Assignment: Section 4.5 Enhanced Assignment x+3 Use​l'Hôpital's Rule to evaluate lim . Then determine the limit using limit laws and commonly known limits. 2 x→ − 3 x − 9 Use​l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. x+3 lim = lim 2 x→ − 3 x − 9 x→ − 3 1 2x Choose the limit equivalent to the given limit that can be evaluated using limit laws and commonly known limits. 1 x A. 3 + x lim x→ − 3 2 9 1− x B. 1 lim x+3 x→ − 3 C. x lim 2 x→ − 3 x D. 1 lim x−3 x→ − 3 2 x+3 The limit found using either method is lim 2 = x→ − 3 x − 9 ​(Simplify your​answer.) − 1 6 . 2. 2 3x − 7x Use​l'Hôpital's Rule to evaluate lim . Then determine the limit using limit laws and commonly known limits. 2 x→∞ 5x + 3 Use​l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. 2 3x − 7x lim = lim 2 x→∞ 5x + 3 x →∞ 6 10 Choose the limit equivalent to the given limit that can be evaluated using limit laws and commonly known limits. 7 3− A. lim x →∞ x 3 5+ 2 x 3x − 7 lim B. x→∞ 5x + 3 x C. 3 7x lim − 5 3 x →∞ D. 3 − 7x lim 5+3 x →∞ 2 3x − 7x The limit found using either method is lim = 2 x→∞ 5x + 3 ​(Simplify your​answer.) 3 5 3. Use​l'Hôpital's Rule to find the limit. 3 t − 2t + 115 lim 2 t→ − 5 t − t − 30 3 t − 2t + 115 lim = 2 t→ − 5 t − t − 30 − 73 ​(Type an exact​answer.) 11 4. Use​l'Hôpital's rule to find the following limit. 3 2x − 4x lim 3 x→∞ 3x + 5 3 2x − 4x lim = 3 x→∞ 3x + 5 2 ​(Type an integer or a​fraction.) 3 5. Use​l'Hôpital's Rule to evaluate lim t →0 lim t →0 5 sin 7t − 4t 5 sin 7t − 4t 3 . 3 = 0 ​(Type an exact​answer.) . 6. Use​l'Hôpital's rule to find the limit. 2 11x lim cos x − 1 x →0 What must be done to put the given expression in a form to which​l'Hôpital's rule can be directly​applied? ∞ A. ​Nothing, the given expression already has the indeterminate form ∞ . B. ​Nothing, the given expression already has the indeterminate form 0 • ∞. 2 The expression must be rewritten as lim 11x ( sec x − 1). x →0 C. D. ​Nothing, the given expression already has the indeterminate form 0 0 . Find the limit. 2 11x lim = cos x − 1 x →0 − 22 ​(Type an exact​answer.) 7. Use​l'Hôpital's Rule to find the limit. 1 − sin θ lim θ→ 2 1 − sin θ lim θ→ π 2 + 2 cos 2θ π 2 + 2 cos 2θ 1 = ​(Type an exact​answer.) 8 2 8. Find the exact value of the following expression for the given value of θ. Do not use a calculator. tan (θ/ 2) if θ = π/ 2 If θ = π/ 2​, then tan (θ/ 2) = 1 . ​(Simplify your answer. Type an exact​answer, using radicals as​needed.) 9. Find the derivative of y with respect to θ of y = 7θ( sin ( ln (7θ)) + cos ( ln (7θ))). dy dθ = 14 cos ( ln (7θ)) 10. Use logarithmic differentiation to find the derivative of y with respect to x. y= 3 x(x − 1) 2 x +9 dy dx = 1 3 x(x − 1) 3 2 x +9 1 x + 1 x−1 − 2x 2 x +9 11. Use​l'Hôpital's Rule to find the limit. y+4 −2 lim y →0 y y+4 −2 lim y →0 1 = y ​(Type an integer or a simplified​fraction.) 4 12. Use​l'Hôpital's rule to find the following limit. lim θ →0 lim θ →0 5 sin (θ) cos (θ) − θ tan (θ) − θ 5 sin (θ) cos (θ) − θ tan (θ) − θ ∞ = ​(Simplify your​answer.) 13. Find the limit. 6 1−x lim x x →1 + 6 lim x x →1 + 1−x 1 = (Type an exact​answer.) e6 14. Find the following limit. lim x x →0 + − 17 / ln x − 17 / ln x lim x x →0 + 1 = ​(Type an exact​answer.) e17 15. Find the limit. 1 lim (1 + 2x) 2 ln x x →∞ 1 lim (1 + 2x) x →∞ 2 ln x = e1 ​(Type an exact​answer.) 16. Find the following limit. lim (6x) + x →0 5x What must be done to put the given expression in a form to which​l'Hôpital's rule can be directly​applied? 1 . A. The expression must be rewritten as lim − 5x x→0+ (6x) ∞ B. ​Nothing, the given expression already has the indeterminate form ∞ . C. ​Nothing, the given expression already has the indeterminate form 0 0 . D. The expression must be rewritten by taking the natural logarithm. Find the limit. lim (6x) x →0 + 5x = 1 ​(Type an exact​answer.) 17. Find the following limit. 4 lim 7 + x ln x x →0 + 4 lim 7 + x ln x = x →0 + 7 ​(Simplify your​answer.) 18. Find the following limit. π lim 2x sec + x →0 π lim 2x sec x →0 + 2 2 −x −x = 2 ​(Simplify your​answer.) 19. ​L'Hôpital's rule does not help with the limit below. Find the limit some other way. lim x →∞ lim x →∞ 36x + 3 x+4 36x + 3 x+4 = 6 ​(Simplify your​answer.) 20. Use limits to find horizontal asymptotes for each function. a. y = x 10 10 tan x a. y = 1 b. y = 0, 5 4 b. y = 5x + e 4x 4x + e 5x ​(Simplify your answer. Use a comma to separate answers as​needed.) ​(Simplify your answer. Use a comma to separate answers as​needed.)