Question

For which of the following limit expressions does L'Hôpital's Rule not apply? Explain your reasoning.

A. limₓ→₀ x²cos(x)/9x
B. limₓ→ −9 (x²−81/x-9)
C. limₓ→₁ x³−1/x-1
D. limₓ→₀ ln(x+1)−xsin(x)

Answer 1

L'Hôpital's Rule does not apply to B: Lim₁→−₉ (x²−81/x-−9) because it does not create an indeterminate form, unlike the other options where the rule could potentially be applied.

Step-by-step explanation:

L'Hôpital's Rule applies to indeterminate forms like 0/0 or ∞/∞. Let's evaluate each limit.

• Lim₁→₀ x²cos(x)/9x: As x approaches 0, the numerator and the denominator both approach 0. L'Hôpital's Rule can be applied.
• Lim₁→−₉ (x²−81/x-−9): The numerator (x² - 81) approaches 64 and the denominator (x + 9) approaches 0. This is not an indeterminate form and hence L'Hôpital's Rule does not apply.
• Lim₁→₁ x³−1/x-1: The numerator and the denominator both approach 0. L'Hôpital's Rule can be applied.
• Lim₁→₀ ln(x+1) −xsin(x): As x approaches 0, the expression ln(x+1) approaches ln (1) which is 0, and xsin(x) also approaches 0; however, since this condition does not yield an indeterminate form of the type 0/0 or ∞/∞, it is unclear without further analysis if L'Hôpital's Rule applies directly.

Therefore, L'Hôpital's Rule does not apply to option B: Lim₁→−₉ (x²−81/x-−9) because it is not an indeterminate form.