Question

What will be the value of \(\lim\limits_{x \rightarrow 1} x^{\frac{1}{1-x}}\)?

(a) 1/e

(b) e

(c) 0

(d) 1

I have been asked this question by my college professor while I was bunking the class.

The doubt is from First Order Derivative in section Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct choice is (a) 1/e

Explanation: We have, y = x^(1/(1 – x))

So, log y = (log x)/(1 – x)

So, \(\lim\limits_{x \rightarrow 1}\)log y = \(\lim\limits_{x \rightarrow 1}\)(log x)/(1 – x)

Now, using L’Hospital’s rule,

\(\lim\limits_{x \rightarrow 1}\)log y = \(\lim\limits_{x \rightarrow 1}\)((\(\frac{1}{x})\) / – 1)  

=>, \(\lim\limits_{x \rightarrow 1}\)log y = -1

So, \(\lim\limits_{x \rightarrow 1} x^{\frac{1}{1-x}}\) = 1/e.