$$ \lim_{x\to 0} (\frac{(5x + 1)^{20} - (20x + 1)^{5}}{\sqrt[5]{1 + 20x^{2}}-1}) $$
Hello! I need to solve this limit. I had solved it with the rule of L'Hôpital, but i can't without it. I tried multiplicatio using Special Limits, but i simplified it only to $$ \lim_{x\to 0} (\frac{e^{100x} - e^{100x}}{e^{4x^2}-1})$$ So i think i had done something wrong and i should do it by another way. Please help me, I must solve it using only Special Limits and simple transformations. I can't use derivatives.
You can just use the Taylor series on the bottom and expand the top, keeping only the first terms that do not cancel. $$(5x + 1)^{20} - (20x + 1)^{5}=1+100x+20\cdot 19 \cdot \frac 12\cdot 5^2 x^2+O(x^3)-1-100x-5\cdot 4 \cdot \frac 12\cdot 20^2+O(x^3)\\=750x^2+O(x^3)\\ \sqrt[5]{1 + 20x^{2}}-1=1+\frac {20x^2}5+O(x^4)-1=4x^2+O(x^4)$$ and the ratio is just $$\frac {750}4+O(x)$$