Question

Show $\underset{x\to 0}{lim}\frac{e^{1/x}-1}{e^{1/x}+1}$ does not exist.

Answer 1

$\underset{x\to 0}{lim}\left(\frac{e^{1/x}-1}{e^{1/x}+1}\right)$

$=\underset{x\to 0}{lim}\left(\frac{1-e^{-1/x}}{1+e^{-1/x}}\right)$ $\{$ Multiplying $e^{-1/x}$ on $N^r$ & $D^r$$\}$.

$=\frac{1-e^{-\infty }}{1+e^{-\infty }}$

$=1$.