The value of `underset(xrarr0)(lim)(1)/(x^(3)) int_(0)^(x)(tln(1+t))/(t^(4)+4) dt`

Options

A. 0
B. `1//12`
C. `1//24`
D. `1//24`

Answer

B

Explanation

`underset(x to 0)lim(1)/(x^(3))(t " In"(1+t))/(t^(4)+4)dt`
`=underset(x to0)lim(1)/(x^(3))(overset(x)underset(0)int (t"In"(1+t))/(t^(4)+4)dt)/(x^(3))`
`=underset(x to 0)lim ((x" In "(1+x))/(x^(4)+4))/(3x^(2))`[Applying De'L' Hospital's rule]
`=underset(x to 0)lim(x" In "(1+x))/(x(x^(4)+4))`
`=(1)/(3)xxunderset(x to 0)("lim")(log(1+x))/(x)xx(1)/(x^(4)+4)=(1)/(3)xx1xx(1)/(4)=(1)/(12)`