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# Prove that the function $f (x) = x^n$ is continuous at $x = n$, where n is a positive integer.

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## 1 Answer

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Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
Given $f(x) = x^n$:
$\Rightarrow \lim\limits_{x\to n} f(x) = \lim\limits_{x\to n} f(x) = x^n = n^n$
$\Rightarrow f(n) = n^n$
Therefore, $f$ is continuous at $x=n$
answered Apr 4, 2013

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