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

Given: $f(x) = {x^n}$ where n is a positive integer.

Continuity at x = n, $\mathop {\lim }\limits_{x \to n} f\left( x \right) = \mathop {\lim }\limits_{x \to n} \left( {{x^n}} \right) = {n^n}$

And $f\left( n \right) = {n^n}$

Since $\mathop {\lim }\limits_{x \to n} f\left( x \right) = f\left( x \right)$, therefore, $f\left( x \right)$ is continuous at x = n.

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