4. Applications of Derivatives

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

105. lim(x→∞) x arctan(2/x)

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

e. x

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

e. x - 2ln(x)

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

g. ln(ln x)

19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

a. x-3

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

a. 10x^4 + 30x + 1

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

c. √(1+x^4)

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

1. lim (x → -2) (x + 2) / (x² - 4)

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

3. lim (x → ∞) (5x² - 3x) / (7x² + 1)

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

27. lim (x → (π/2)^-) (x - π/2) sec x

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

30. lim (θ → 0) ((1/2)^θ - 1) / θ

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

32. lim (x → 0) (3^x - 1) / (2^x - 1)