# Use L'Hospital Rule to evaluate the following limits. lim_{xrightarrow0}(tanh x)^x

Question
Limits and continuity
Use L'Hospital Rule to evaluate the following limits.
$$\lim_{x\rightarrow0}(\tanh x)^x$$

2020-12-01
Consider the given expression,
$$\lim_{x\rightarrow0}(\tanh x)^x$$
Since the function does not have any denominator hence, the L'Hospital rule cannot be applied.
Thus,
$$\lim_{x\rightarrow0^+}(\tanh x)^x=\lim_{x\rightarrow0^+}(\frac{e^x-e^{-x}}{e^x+e^{-x}})^x$$
$$=(\frac{e^0-e^{-0}}{e^0+e^{-0}})^0$$
$$=1$$
Which is the required value.

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