4. Applications of Derivatives

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))

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

91. lim(x→π/2⁻) (sec(7x))(cos(3x))

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

93. lim(x→0) (csc(x) - cot(x))

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

97. lim(x→0) (10^x - 1)/x

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

99. lim(x→0) (2^sin(x) - 1)/(e^x - 1)

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

102. lim(x→0) (x sin(x²))/(tan³x)

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

104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)

89. Use limits to find horizontal asymptotes for each function.

a. y = x tan(1/x)