Question

The value of $k$ which makes $\text{f(x)=}\left\{\sin \left(\frac{1}{\text{x}}\right)\text{,x≠0}\right\}\text{(k,x=0)}$ continuous at $\text{x=0}$ is

• A. $8$
• B. $1$
• C. $-1$
• D. none of these

Answer 1

The correct option is D

none of these

Finding the value of $k$ :

Let $\text{g(x)=}\sin \left(\text{x}\right)\text{,h(x)=}\frac{1}{x}$ then $\text{f(x)=goh(x)=g(h(x)}$

If $\text{g(x)}$ and $\text{h(x)}$ are both continuous at $\text{x=0}$, then $\text{goh(x)}$ is continuous at $\text{x=0}$ so $\text{f(x)=goh(x)}$ is continuous at $\text{x=0}$ .

But here we see that $\text{h(x)=}\frac{1}{\text{x}}$ is not defined at $\text{x=0}$ so not continuous at $\text{x=0}$ .

Therefore $\text{f(x)=}\sin \left(\frac{1}{\text{x}}\right)$ is not continuous at $\text{x=0}$ for any value of $\text{k}$.

Hence option (D) is the correct answer .