5. Problem: Evaluate $\lim _{x \rightarrow 1} x^{\frac{x}{\sin (1-x)}}$.
Solution: Rewrite the expression to evaluate as $e^{\ln x^{\sin (1-x)}}$. Then we must evaluate $\lim _{x \rightarrow 1} e^{\ln x^{\sin (1-x)}}$. $\lim _{x \rightarrow 1} \ln x^{\frac{x}{\sin (1-x)}}=\lim _{x \rightarrow 1}\left(\frac{x}{\sin (1-x)} \ln x\right)$. Because direct calculation of the limit results in indeterminate form $\left(\frac{1}{0} \cdot 0\right)$, we can use L'Hopital's rule to evaluate the limit. By L'Hopital's rule, $\lim _{x \rightarrow 1}\left(\frac{x}{\sin (1-x)} \ln x\right)=$ $\lim _{x \rightarrow 1} \frac{\ln x+1}{-\cos (1-x)}$. This limit is simply -1 .
Hence $\lim _{x \rightarrow 1} e^{\ln x^{\frac{x}{\sin (1-x)}}}=e^{-1}=\frac{1}{e}$.