4. Applications of Derivatives

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)

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

b. f(x) = x, g(x) = x², (a, b) arbitrary

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

55. lim (x → ∞) (ln x)^(1/x)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

60. lim (x → 0) (e^x + x)^(1/x)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

63. lim (x → ∞) ((x + 2)/(x - 1))^x

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

66. lim (x → 0⁺) x (ln x)²