# Limit at Infinity of x^n

## Theorem

Let $x \mapsto x^n$, $n \in \R$ be a real function which is continuous on the open interval $\openint 1 {+\infty}$.

Let $n > 0$.

Then $x^n \to +\infty$ as $x \to +\infty$.

## Proof

$\forall n > 0: n \ln x < x^n$

which, by Multiple Rule for Continuous Functions, implies:

 $\displaystyle \lim_{x \mathop \to +\infty} n \ln x$ $=$ $\displaystyle n \lim_{x \mathop \to +\infty} \ln x$
$n \ln x \to +\infty$ as $x \to +\infty$

The result follows from the Push Theorem.

$\blacksquare$