Question

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

Answer 1

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