How to compute $\lim_{x \to 1}x^{\frac{1}{1-x}}$? I need to compute $$\lim_{x \to 1}x^{\frac{1}{1-x}}$$
The problem for me is negative $x$.

Set $x-1=h$
So, we have $$\lim_{h\to0}(1+h)^{-\frac1h}=\left([1+h]^{\frac1h}\right)^{-1}=e^{-1}$$

Alternatively, if $A=\lim_{x\to1}x^{\dfrac1{1-x}},$
$\ln A=\lim_{x\to1}\dfrac{\ln\{1-(1-x)\}}{1-x}$
Setting $x-1=h,$
$\ln(A)=-\lim_{h\to0}\dfrac{\ln(1+h)}h=-1$
