Evaluate $\lim\limits_{x \to 0} \dfrac{(ax+1)^b - (bx+1)^a}{x^2}$

Question

Evaluate $$\lim\limits_{x \to 0} \dfrac{(ax+1)^b - (bx+1)^a}{x^2}$$ where $a, b$ are non-negative integers.

I solved this with the help of L'Hopital's rule, but still I'm looking for a different solution (i.e no L'Hopital's rule). Thanks in advance.

Answer 1

Hint: Using the binomial theorem, you have the following approximations $$(1+ax)^b \approx 1 + abx + \frac{b(b-1)}{2}(ax)^2$$ $$(1+bx)^a \approx 1 + abx + \frac{a(a-1)}{2}(bx)^2$$ when $x \to 0$. The rest is straightforward.