Question:
Prove that the function $f(x)=x^{n}$ is continuous at $x=n$, where $n$ is a positive integer.
Solution:
The given function is $f(x)=x^{n}$
It is evident that $f$ is defined at all positive integers, $n$, and its value at $n$ is $n^{n}$.
Then, $\lim _{x \rightarrow n} f(n)=\lim _{x \rightarrow n}\left(x^{n}\right)=n^{n}$
$\therefore \lim _{x \rightarrow n} f(x)=f(n)$
Therefore, f is continuous at n, where n is a positive integer
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