Prove that the function $f(x) = x^{n}$ is continuous at $x = n$,where $n$ is a positive integer.
(N/A) The given function is $f(x) = x^{n}$.
It is evident that $f$ is defined at all positive integers $n$,and its value at $x = n$ is $f(n) = n^{n}$.
Now,we evaluate the limit of the function as $x$ approaches $n$:
$\lim_{x \to n} f(x) = \lim_{x \to n} (x^{n}) = n^{n}$.
Since $\lim_{x \to n} f(x) = f(n) = n^{n}$,the condition for continuity is satisfied.
Therefore,the function $f(x) = x^{n}$ is continuous at $x = n$,where $n$ is a positive integer.
It is evident that $f$ is defined at all positive integers $n$,and its value at $x = n$ is $f(n) = n^{n}$.
Now,we evaluate the limit of the function as $x$ approaches $n$:
$\lim_{x \to n} f(x) = \lim_{x \to n} (x^{n}) = n^{n}$.
Since $\lim_{x \to n} f(x) = f(n) = n^{n}$,the condition for continuity is satisfied.
Therefore,the function $f(x) = x^{n}$ is continuous at $x = n$,where $n$ is a positive integer.
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