Question

lf $f\left(x\right)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2{)}^2}+\dots +\frac{x^2}{(1+x^2{)}^n}+\dots$

then at $x=0$ which of the following is correct?

• 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

Answer: (D)

$f\left(x\right)$ is differentiable

Since the given series is a Infinite GP with a common ratio,
$r=\frac{1}{1+x^2}$
Hence,
$f\left(x\right)=\frac{x^2}{1-\frac{1}{1+x^2}}$
$f\left(x\right)=1+x^2$ which is continuous at $x=0$
$f^{\prime }\left(x\right)=2x$ which is continuous at $x=0$
Hence, $f\left(x\right)$ is differentiable at $x=0$