I am not able to get that is it expansion or differentiaition

The question is asking to find limit of the given function. We will apply L hospital rule here i.e. we will differentiate numerator and denominator.

For differentiating the integral we will use theorem, diff of integral f(x).g(x) = f(g(x))∗g(x)′−f(h(x))∗h(x)′ .

So the given limit=lim x->0 (x*ln(1+x))/((x^4+4)*3x^2)

=lim x->0 1/3 * ln(1+x) /(x(x^4 +4))

Now putting value of x as 0,we get

Ans=1/12. So option (2) 1/12 is the correct answer.