4. Applications of Derivatives

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.

lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

More limits Evaluate the following limits.

lim_x→1 (x ln x - x + 1) / (xln²x)

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x² ln x; x³

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Exponential growth rates

a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Exponential growth rates

b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ ln ((x +1) / (x-1))

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→π / 2- (sin x) ^tan x

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_ x→0 ⁺ | ln x | ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ (1 - (3/x))ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 ( x- 1)^sinπx

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 2x + 1

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = sin² x