4. Applications of Derivatives

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

lim_x→0 (1 - cosⁿ x) / x²

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

lim_u→ π/4 (tan u - cot u) / (u - π/4)

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

lim_z→0 (tan 4z) / (tan 7z)

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

lim_x→ 0 (1 - cos 3x) / 8x²

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

lim_x→ 0 (sin² 3x) / x²

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

lim_x→π (cos x +1 ) / (x - π )²

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.

lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

More limits Evaluate the following limits.

lim_x→1 (x ln x - x + 1) / (xln²x)

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x² ln x; x³

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Exponential growth rates

a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Exponential growth rates

b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

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

lim_x→∞ ln ((x +1) / (x-1))