Let $f_n(x)=\dfrac{\sin(nx)}{1+n^2x^2}$.
First note that $f_n(x) \to 0$ pointwise. Also, if you fix $\epsilon = \dfrac{1}{4}$ and let $x=\dfrac{1}{n}$, then $|f_n(\frac{1}{n})-f(x)|=| \frac{\sin(1)}{2}-0|=|\frac{\sin(1)}{2}|\geq \frac{1}{4}.$
My question is, how much more should I show? That is, I'm not sure how to make this argument more clearer. I was marked off for lack of details.