<s>[INST] Find $\lim\limits_{x\to 1}\frac9{1-x^9}-\frac7{1-x^7}$ without using L'Hopital $$\lim_{x\to 1}\frac9{1-x^9}-\frac7{1-x^7}$$
I did this with l'Hôpital's rule, but how can we do this problem other than that? Any hints will be good.
 [/INST] The difference of fractions equals $\frac{9(1-x^7) - 7(1-x^9)}{(1-x^7)(1-x^9)}$.
Try to divide out the common $(1-x)^2$ terms in both the bottom and the top.
In the bottom we get $(1+x+x^2+\ldots+x^8)(1+x + x^2 + \ldots + x^6)$
In the top $7x^7 + 14x^6 + 12x^5 + 10x^4 + 8x^3 + 6x^2 + 4x + 2$ (multiply by $(1-x)^2$ to check both). Now we can just substitute $x=1$ in both. I get 63 twice OTTOMH.
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