Question

$f\left(x\right)=x^2(1-\cos \left(\frac{2}{x}\right))$ for $x\ne 0$ and $f\left(0\right)=k$. If $f\left(x\right)$ is continuos at $x=0$ then find $k$.

Answer 1

$f\left(x\right)=x^2(1-\cos \left(\frac{2}{x}\right))=x^2\times 2{\sin }^2\left(\frac{2}{2x}\right)=2x^2{\sin }^2\left(\frac{1}{x}\right)$

$f\left(x\right)=\frac{2{\sin }^2\left(\frac{1}{x}\right)}{{\left(\frac{1}{x}\right)}^2}$

So, $\underset{x\to 0}{lim}f\left(x\right)=2=k$

So, $k=2.$