4. Applications of Derivatives

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_t→0 (1 - cos 6t) / 2t

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_Θ→0 (3 sin² 2Θ) / Θ²

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_Θ→0 2Θ cot 3Θ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_y→0⁺ (ln¹⁰ y) / √y

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ x³ (1/x - sin 1/x)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ x²ˣ

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 (x + cos x)¹/ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→0 csc x sin⁻¹ x

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

x¹⸍² and x¹⸍³

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

eˣ and 3ˣ

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

2ˣ and 4ˣ⸍²

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cos xⁿ) / x²ⁿ