4. Applications of Derivatives

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

lim_x→ e (ln x - 1) / (x - 1)

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

lim_x→ 0⁺ (1 - ln x) / (1 + ln x)

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

lim_x→ 0 (3 sin 4x) / 5x

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

lim_x→2π (x sin x + x² - 4π²) / (x - 2π)

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

lim_x→ 0 (eˣ - x - 1) / 5x²

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

lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))

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

lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

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

lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

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

lim_x→0 (sin x - x) / 7x³

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

lim_x→∞ (e¹/ₓ - 1)/(1/x)

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

lim_x→∞ (tan⁻¹ x - π/2)/(1/x)

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

lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)

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

lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)

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

lim_v→3 (v-1-√(v²-5)) / (v-3)

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

lim_y→2 (y²+y-6) / (√(8-y²)-y)