How to show $\lim_{x\to 1}2^{1/(x-1)} $ does not exist? How to show $$\lim \limits_{x\to 1}2^{\frac{1}{x-1}} $$ doesn't exist?


I understand that we have to show the left-hand limit is not equal to the right-hand limit. But how to do that?
Let $f(x)=2^{\frac{1}{x-1}}$ and $x_n:=1+1/n$. Then

$f(x_n)=2^n \to \infty$ for $n \to \infty$.

If $y_n=1-1/n$, then $f(y_n)=1/2^n \to 0$ for $n \to \infty$.
$$2^{1/(1+\epsilon-1)}$$ is unbounded for $\epsilon&gt;0$. No limit.