Question

L’Hˆopital’s Rule limit calculations: Calculate each of the

limits that follow. Some of these limits are easier to calculate

by using L’Hopital’s rule, and some are not.

$\underset{x\to 1^{+}}{\lim }{x}^{1/(x-1)}$

Short answer

Limit is $e$and it is easy to find using L'Hospital rule

Step-by-step solution

Step 1. Given information

An expression is given as$\underset{x\to 1^{+}}{\lim }{x}^{1/(x-1)}$

Step 2. Evaluating limit

Use properties of limits,

$$\begin{gathered} \underset{x\to 1^{+}}{\lim }\ln x^{\left(\frac{1}{x-1}\right)}=\underset{x\to 1^{+}}{\lim }\left(\frac{1}{x-1}\right)\ln x \\ =\underset{x\to 1^{+}}{\lim }\left(\frac{\ln x}{x-1}\right) \\ =\underset{x\to 1^{+}}{\lim }\left(\frac{\frac{1}{x}}{1-0}\right) \\ =\underset{x\to 1^{+}}{\lim }\frac{1}{x} \\ =\frac{1}{1} \\ =1 \\ \Rightarrow \underset{x\to 1^{+}}{\lim }{x}^{\left(\frac{1}{x-1}\right)}=e^1=e \end{gathered}$$