<s>[INST] 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
 [/INST] 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
$$
f(x)=
\begin{cases}
1+x^2 &\text{if $x\ne0$}\\
0&\text{if $x=0$}
\end{cases}
$$
</s>