Question

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

Answer 1

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,
$\mathop {\lim }\limits_{x \to a} f\left( x \right) \ne f\left( a \right)$