4. Applications of Derivatives

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)

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

9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)

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

12. lim (x → ∞) (x - 8x²) / (12x² + 5x)

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

14. lim (t → 0) sin 5t / 2t

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

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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

19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)

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

22. lim (x → 1) (x - 1) / (ln x - sin πx)

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

85. lim(x→1) (x² + 3x - 4)/(x - 1)

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

88. lim(x→0) (tan x)/(x + sin(x))