# What's the Limit of x^(1/lnx) as x approaches zero from the right?

May 10, 2018

$e$

#### Explanation:

Since ${x}^{y} = \exp \left(y \ln \left(x\right)\right)$ for all $x , y$, we can rewrite the expression as

${x}^{\frac{1}{\ln \left(x\right)}} = \exp \left(\frac{1}{\ln \left(x\right)} \ln \left(x\right)\right) = e$

$f : \left(0 , \infty\right) \to \mathbb{R} : f \left(x\right) = e$
since it has constant value $e$ in all the intervals $\left(0 , \epsilon\right)$, its limit on ${0}^{+}$ is $e$