Question

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}

Answer 1

`= 1/3`

Explanation:

`lim_( x to 0) ( int_0^x (t^2/(t^4+1))dt)/x^3`
This is `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) ( d/dx int_0^x (t^2/(t^4+1))dt)/(d/dx (x^3 ) )`

`= lim_( x to 0) ( x^2/(x^4+1))/(3 x^2)`

`=1/3 lim_( x to 0) 1/(x^4+1) = 1/3`