Question

If $f\left(x\right)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{{(1+x^2)}^2}+...+\frac{x^2}{{(1+x^2)}^n}+....$ then at $x=0$

• A. $f\left(x\right)$ has no limit
• B. $f\left(x\right)$ is discontinuous
• C. $f\left(x\right)$ is continuous but not differentiable
• D. $f\left(x\right)$ is differentiable

Answer 1

The correct option is B $f\left(x\right)$ is discontinuous

For $x=0,f\left(0\right)=0$ and for $x\ne 0$

$f\left(x\right)=x^2[1+\frac{1}{1+x^2}+\frac{1}{{(1+x^2)}^2}+...]$
$f\left(x\right)=x^2\frac{1}{1-\frac{1}{\left({1+x}^2\right)}}=\frac{x^2(1+x^2)}{x^2}=1+x^2$
Since ${lim}_{x\to 0}f\left(x\right)=1$, so f is not continuous at $x=0$

And hence non-differentiable at $x=0$