88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → 0⁺ (tanh x)ˣ

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Recognize that the limit is of the form $\lim_{x \to 0^+} (\tanh x)^x$. Since $\tanh x$ approaches 0 as $x$ approaches 0, and the exponent $x$ also approaches 0, this is an indeterminate form of type \$0^0$.

Rewrite the expression using the exponential and logarithm functions to handle the indeterminate form: $\lim_{x \to 0^+} (\tanh x)^x = \lim_{x \to 0^+} e^{x \ln(\tanh x)}$.

Focus on evaluating the exponent limit: $\lim_{x \to 0^+} x \ln(\tanh x)$. This is a product of $x$ and $\ln(\tanh x)$, which tends to $0 \cdot (-\infty)$, another indeterminate form.

Rewrite the product as a quotient to apply l'Hôpital's Rule: $\lim_{x \to 0^+} \frac{\ln(\tanh x)}{1/x}$. Now, as $x \to 0^+$, the numerator tends to $-\infty$ and the denominator tends to $+\infty$, so the limit is of the form $\frac{-\infty}{\infty}$.

Apply l'Hôpital's Rule by differentiating numerator and denominator with respect to $x$: differentiate $\ln(\tanh x)$ and $1/x$, then take the limit of their ratio as $x \to 0^+$. This will give the limit of the exponent, which you can then use to find the original limit.

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Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially at points where direct substitution leads to indeterminate forms, is essential for analyzing function behavior near those points.

Indeterminate Forms

Indeterminate forms like 0^0, ∞/∞, or 0/0 occur when direct substitution in a limit does not yield a clear value. Recognizing these forms is crucial because they signal the need for special techniques, such as algebraic manipulation or applying l’Hôpital’s Rule, to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s Rule helps evaluate limits that result in indeterminate forms 0/0 or ∞/∞ by differentiating the numerator and denominator separately. Applying this rule correctly simplifies complex limits, making it easier to find the limit’s value when direct substitution fails.

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.lim x → 0⁺ (tanh x)ˣ