Evaluate $\lim\limits_{x\to 0}\frac{1}{x^3}\int_0^x\frac{t^2}{t^4+1}dt$ . $$\lim\limits_{x\to 0}\frac{1}{x^3}\int_0^x\frac{t^2}{t^4+1}dt$$
To determine the above value, do I need to integrate first? Or, is there some other way to do this?

Let $f(x)=\int_0^x\dfrac{t^2}{t^4+1}dt$
Now you have to compute $\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}$.
This can be done in a hospital !
