Question

If $f\left(x\right)=\{\begin{array}{ccc}{x}^a\cos \left(1/x\right),& if& x\ne 0\\ 0,& if& x=0\end{array}$ is continuous at $x=0$ then

• A. $\alpha <0$
• B. $\alpha >0$
• C. $\alpha =0$
• D. $\alpha \ge 0$

Answer 1

The correct option is B $\alpha >0$

$f\left(x\right)=\left\{\begin{array}{ccc}{x}^a\cos \left(1_{/x}\right)& if& x\ne 0\\ 0& if& x=0\end{array}\right\}$

Continuous at $x=0$, then

$\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=0$

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}{x}^a\cos \left(1_{/x}\right)$

If $a\le 0$

$\underset{x\to 0}{lim}f\left(x\right)$ does not exist

If $a>0$

$\underset{x\to 0}{lim}f\left(x\right)=0$

So, $a>0$ for $f\left(x\right)$ to be continuous at $x=0$.