Question

By the definition of continuity, how do you show that \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\]?

Answer 1

Hint: From the question given, we have been asked to show \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\] by using the definition of continuity. By using the definition of continuity, to show that \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\], first of all, we have to know the definition of continuity.

Complete step by step solution:
For answering this question we need to show that \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at $x=0$
First we have to know about the definition of continuity. Definition of continuity is shown below.
Definition of continuity: \[f\] is continuous at \[a\] , if and only if \[\displaystyle \lim_{x \to a}f\left( x \right)=f\left( a \right)\]
Irrespective of the path in which $x$ tends to $a$ .
If \[f\left( a \right)\] does not exist, then \[f\] is not continuous at \[a\].
Therefore, the above written condition describes the definition of continuity.
By using the above written definition of continuity, we have to show that \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\].
First of all, let us find out \[f\left( a \right)\] because if \[f\left( a \right)\] exists then only it will be continuous and if \[f\left( a \right)\] will not exists then the given function is not continuous.
From the question given, we have been asked to show \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\].
Therefore, here \[a=0\]
Now,
\[\Rightarrow f\left( a \right)=a\sin \left( \dfrac{1}{a} \right)\]
\[\Rightarrow f\left( 0 \right)=0\sin \left( \dfrac{1}{0} \right)\]
Therefore, we can clearly observe that \[f\left( a \right)\] is equal to zero.
Now we need to show that the expression $\displaystyle \lim_{x \to 0}x\sin \dfrac{1}{x}=0$ is valid for doing that let us assume $\dfrac{1}{x}=z$ then we can write this expression simply as $\displaystyle \lim_{z\to \infty }\dfrac{1}{z}\sin z=0$ .
From the basic concepts as we know that $\displaystyle \lim_{\theta \to \infty }\dfrac{\sin \theta }{\theta }=0$ we will use that formulae here.
After using that formula here we will have $\displaystyle \lim_{z\to \infty }\dfrac{1}{z}\sin z=0$

Therefore, \[x\sin \left( \dfrac{1}{x} \right)\] is continuous at \[x=0\].

Note: We should be well known about the definition of continuity. We should be very careful while checking the continuity. Also, we should be very careful while checking \[f\left( a \right)\]. Also, we should be well aware of the usage of the definition of continuity. Also, we should be very careful while writing the definition of continuity because the whole problem is dependent on the definition of continuity. Similarly we can verify the continuity of $x\cos \left( \dfrac{1}{x} \right)$ or any other function.