Prove that the function is continuous at x = n, where n is a positive integer.
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, $\lim _{x \rightarrow n} f(n)=\lim _{x \rightarrow n}\left(x^{n}\right)=n^{n}$

$\therefore \lim _{x \rightarrow n} f(x)=f(n)$

Thereforeis continuous at n, where n is a positive integer

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