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
From Upper Bound of Natural Logarithm:


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

which, by Multiple Rule for Continuous Real Functions, implies:

From Logarithm Tends to Infinity:


 * $n \ln x \to +\infty$ as $x \to +\infty$

The result follows from the Push Theorem.

Also see

 * Sequence of Powers of Reciprocals is Null Sequence
 * Limit at Infinity of Identity Function