Question

The value of `lim_(x rightarrow 0) 1/x^3 int_0^x (t log (1 + t))/(t^4 + 4) dt` is ______.

Options

• 0

• `1/12`

• `1/24`

• `1/64`

Answer 1

The value of `lim_(x rightarrow 0) 1/x^3 int_0^x (t log (1 + t))/(t^4 + 4) dt` is `underlinebb(1/12)`.

Explanation:

Given, `lim_(x rightarrow 0) 1/x^3 int_0^x (t log (1 + t))/(t^4 + 4) dt`

`\implies lim_(x rightarrow 0) (int_0^x (t log (1  +  t))/(t^4  +  4) dt)/x^3`

Using L' Hospital's rule, we get

`lim_(x rightarrow 0) ((x log (1  +  x))/(x^4  +  4))/(3x^2)`

= `lim_(x rightarrow 0) (log(1 + x))/(3x) . 1/(x^4 + 4)`

= `1/3 . 1/4`

= `1/12`