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**Hint:**Here first we will calculate the value of the function when the limit of x tends to n and also the value of the given function at \[x = n\] and then apply the condition for continuity of function to prove the given statement.

The condition for continuity of a function f(x) at \[x = a\] is:

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]

**Complete step-by-step answer:**

Let us first consider the given function:-

\[f\left( x \right) = {x^n}\;\;\]

Now we know that the condition for continuity of a function f(x) at \[x = a\] is:

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]

And since we have to prove the given function is continuous at \[x = n\]

Hence we have to show:

\[\mathop {\lim }\limits_{x \to n} f\left( x \right) = f\left( n \right)\] by the definition of continuity of a function.

Let us now consider the Left hand side:-

\[LHS = \mathop {\lim }\limits_{x \to n} f\left( x \right)\]

Putting the value of f(x) we get:-

\[LHS = \mathop {\lim }\limits_{x \to n} {x^n}\]

Now evaluating the limit we get:-

\[

LHS = {\left( n \right)^n} \\

\Rightarrow LHS = {n^n} \\

\]

Now let us consider Right hand side:-

\[RHS = f\left( n \right)\]

Putting \[x = n\] in the function we get:-

\[

RHS = {\left( n \right)^n} \\

\Rightarrow RHS = {n^n} \\

\]

Now since \[LHS = RHS\]

Therefore the function is continuous at \[x = n\]

Hence proved.

**Note:**Students should take a note that a function f(x) is continuous only if \[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]

Also, all functions which are continuous in an interval have continuous graph i.e, the graph does not break at any point in that interval.

A function is said to be discontinuous in an interval if its graph breaks at some points in that interval and also, mathematically,

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) \ne f\left( a \right)\]

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Prove that the function \[f\left( x \right) = {x^n}\;\;\] is continuous at \[x = n\], where n is a positive integer.

Class 12 MATHS | Continuity & Differentiability | NCERT EXERCISE 5 .1 (Question - 4) | Abhishek Sir

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