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:

\(\ds \lim_{x \mathop \to +\infty} n \ln x\)

\(=\)

\(\ds n \lim_{x \mathop \to +\infty} \ln x\)

From Logarithm Tends to Infinity:

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

The result follows from the Push Theorem.

$\blacksquare$

Also see

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