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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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  • 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 by balaji.thirumalai

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