<s>[INST] Confused on how to solve $\lim_{x\to 0} \frac{(1+mx)^n-(1+nx)^m}{x^2}$ and $\lim_{x\to 1} (\frac{m}{1-x^m}-\frac{n}{1-x^n})$ I'm having trouble solving these two problems using only elementary math. Find:
$$\lim_{x\to 0} \frac{(1+mx)^n-(1+nx)^m}{x^2}$$
and
$$\lim_{x\to 1} (\frac{m}{1-x^m}-\frac{n}{1-x^n})$$
My confusion may be due to the fact that I don't remember encountering these type of problems before. Do any of you recommend books that can help me improve my problem solving skills via elementary math or on different methods of problem solving (like the two I posted)? 
 [/INST] Expanding the numerator by Binomial theorem, we get
\begin{align*}
(1+mx)^n-(1+nx)^m &= \left(1+ nmx + \binom{n}{2}m^2x^2 + \cdots \right) - \left(1+ mnx + \binom{m}{2}n^2x^2 + \cdots \right) \\
&= \frac{nm(n-m)}{2}x^2 + \cdots
\end{align*}
Hence the first limit is $\frac{nm(n-m)}{2}$
For the second limit, we need
\begin{align*}
\lim_{x \rightarrow 1} \frac{m(1-x^n)-n(1-x^m)}{(1-x^n)(1-x^m)} &= \lim_{x\rightarrow 1} \frac{(1-x)(m(1+x+\cdots +x^{n-1}) - n(1+x+\cdots+x^{m-1}))}{(1-x)^2(1+x+\cdots+x^{m-1})(1+x+\cdots+x^{n-1})}\\
&= \lim_{x \rightarrow 1}  \frac{(m(1+x+\cdots +x^{n-1}) - n(1+x+\cdots+x^{m-1}))}{(1-x)(1+x+\cdots+x^{m-1})(1+x+\cdots+x^{n-1})}\\
&= \frac{1}{mn}\lim_{x \rightarrow 1}  \frac{(m(1+x+\cdots +x^{n-1}) - n(1+x+\cdots+x^{m-1}))}{(1-x)}\\
&= \frac{1}{mn}\lim_{x \rightarrow 1}\frac{m[(1-x)(-x^{n-2}-2x^{n-3} - \cdots - (n-1)) + n] - n[(1-x)(-x^{m-2}-2x^{m-3} - \cdots - (m-1)) + m]}{1-x} \\
&= \frac{1}{mn}\left[m\left(-\frac{n(n-1)}{2}+n\right) - n\left(-\frac{m(m-1)}{2}+m\right) \right]\\
&=\frac{m-n}{2}
\end{align*}
In the above, we have used this:
$$1+x+x^2+\cdots +x^{n-1} = (1-x)(-x^{n-2}-2x^{n-3}-3x^{n-4} - \cdots - (n-2)x - (n-1)) + n $$
obtained by the usual division of $1+x+ \cdots +x^{n-1}$ by $1-x$.
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