# How do you find the lim & definite integrals ? for :{lim as x approaches 0 (1/x^3) * integrals (t^2/t^4+1)dt} intervals{0,x}

May 14, 2018

$= \frac{1}{3}$

#### Explanation:

${\lim}_{x \to 0} \frac{{\int}_{0}^{x} \left({t}^{2} / \left({t}^{4} + 1\right)\right) \mathrm{dt}}{x} ^ 3$
This is $\frac{0}{0}$ indeterminate and made for L'Hopital's rule -- the numerator can be unraveled swiftly by differentiating.

Applying L'H:

$= {\lim}_{x \to 0} \frac{\frac{d}{\mathrm{dx}} {\int}_{0}^{x} \left({t}^{2} / \left({t}^{4} + 1\right)\right) \mathrm{dt}}{\frac{d}{\mathrm{dx}} \left({x}^{3}\right)}$

$= {\lim}_{x \to 0} \frac{{x}^{2} / \left({x}^{4} + 1\right)}{3 {x}^{2}}$

$= \frac{1}{3} {\lim}_{x \to 0} \frac{1}{{x}^{4} + 1} = \frac{1}{3}$