4. Applications of Derivatives

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)

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

35. lim (x → 0⁺) ln(x² + 2x) / ln x

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

37. lim (y → 0) (√(5y + 25) - 5) / y

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

47. lim (t → ∞) (e^t + t²) / (e^t - t)

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

49. lim (x → 0) (x - sin x) / (x tan x)

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

51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

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

5. lim (x → 0) (1 - cos x) / x²

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

7. lim (x → 2) (x - 2) / (x² - 4)