Limit at Infinity of x^n
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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.
$\blacksquare$