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

Question

Use L'Hospital Rule to evaluate the following limits.
$lim_{x \to 0} (\tan h x)^{x}$

Answer 1

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