Question

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

Answer 1

The given function is $f\left(x\right)=x^n$
It is evident that $f$ is defined at all positive integers, $n$ and its value at $n$ is $n^n$.
Now , $\underset{x\to n}{lim}f\left(n\right)=\underset{x\to n}{lim}\left(x^n\right)=n^n=f\left(n\right)$
Therefore, $f$ is continuous at $n$, where $n$ is a positive integer.