Prove that the function f(x) = x^n is continuous at x = n, where … - Vidyadip

Question

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

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Answer

Here $f(x) = x^n,$ Where is a possitive integer.
$^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{n}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{n}}(\text{x}^\text{n}) = \text{n}^\text{n}$
Now $f$ is defined at $x = n$
and $f(n) = n^n$
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{n}}\text{f(x)} = \text{f(n)}$
$\therefore f$ is continous at $x = n.$

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