Examine the continuity of $f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+...+ \frac{x^2}{(1+x^2)^n}+....$ at $x=0$ 
Examine the continuity of $$f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+...+ \frac{x^2}{(1+x^2)^n}+....$$ at $x=0$

I tried to solve the problem using $$\lim_{x \to0^+}f(x)=\lim_{x \to0^-}f(x)=f(0)$$ and I got both limits to be $0$. I think there is an error somewhere since my book says it is discontinuous.
Can you help me out?
 Thankyou

If the dots imply an infinite series, then you can first consider
$$
g(x)=\sum_{n\ge0}\frac{1}{(1+x^2)^n}
$$
that converges to
$$
g(x)=\frac{1}{1-\dfrac{1}{1+x^2}}=\frac{1+x^2}{x^2}
$$
whenever
$$
\frac{1}{1+x^2}<1
$$
that is, for all $x\ne0$.
Thus your function is
$$
f(x)=
\begin{cases}
1+x^2 &\text{if $x\ne0$}\\
0&\text{if $x=0$}
\end{cases}
$$
