Compute the limit of the given function

by RoRi

December 25, 2015

Evaluate the limit.

$\lim_{x \to 0} \frac{e^{-\frac{1}{x^2}}}{x^{1000}}.$

Letting $t = \frac{1}{x^2}$ we have

$\frac{e^{-\frac{1}{x^2}}}{x^{1000}} = \frac{t^{500}}{e^t}.$

Since $t \to +\infty$ as $x \to 0$ we then have

$\lim_{x \to 0} \frac{e^{-\frac{1}{x^2}}}{x^{1000}} = \lim_{t \to +\infty} \frac{t^{500}}{e^t} = 0.$

The final equality follows from Theorem 7.11 (page 301 of Apostol).

Stumbling Robot