Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.41. lim (x → 0⁺) (ln x)² / ln(sin x)

Accepted Answer

First, identify the form of the limit as $$ x 	o 0^+ $$ for the expression $$ \frac{(\ln x)^2}{\ln(\sin x)} . $$Substitute values close to 0 from the right to see if it results in an indeterminate form like $$ \frac{0}{0}  or$$ $$ \frac{\infty}{\infty} . $$Rewrite the limit to understand the behavior of numerator and denominator separately: $$ (\ln x)^2 $$ tends to $$ \infty $$ negatively squared (so tends to $$ +\infty $$), and $$ \ln(\sin x)  as$$ $$ x 	o 0^+ $$ tends to $$ \ln(0) = -\infty . So $$the limit is of the form $$ \frac{\infty}{-\infty} $$, which is an indeterminate form suitable for l’Hôpital’s Rule. Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to $$ x $$: 
- Derivative of numerator: $$ \frac{d}{dx} (\ln x)^2 = 2 \ln x \cdot \frac{1}{x} = \frac{2 \ln x}{x} 
- $$Derivative of denominator: $$ \frac{d}{dx} \ln(\sin x) = \frac{1}{\sin x} \cdot \cos x = \cot x $$ Rewrite the limit using these derivatives: 
$$ \lim_{x 	o 0^+} \frac{\frac{2 \ln x}{x}}{\cot x} = \lim_{x 	o 0^+} \frac{2 \ln x}{x \cot x} $$
Simplify the expression inside the limit if possible, for example by expressing $$ \cot x = \frac{\cos x}{\sin x} . $$Evaluate the new limit by analyzing the behavior of each component as $$ x 	o 0^+ . If $$the limit is still indeterminate, consider applying l’Hôpital’s Rule again or use series expansions to find the limit.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
41. lim (x → 0⁺) (ln x)² / ln(sin x)

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Key Concepts

l’Hôpital’s Rule

Behavior of Logarithmic Functions Near Zero

Limit of Trigonometric Functions Near Zero