Question

Find lim x->0 [(integrating t^3/(1+t^6)dt from 0 to x^2)/x^8].

Answer 1

Looks like a rather unreasonable integral.
Here is what Wolfram says

http://integrals.wolfram.com/index.jsp?expr=x%5E3%2F%281%2Bx%5E6%29&random=false

You also appear to have a typo for the upper end of your integral