Recently my discrete mathematics lecturer gave the following problem under the topic of binomial expansion:
If x is nearly equal to unity (1), prove that:
$$\frac {mx^n - nx^m}{x^n - x^m} = \frac 1{1-x}$$
What I've so far determined that $\vert x \vert < 1$ since the denominator is a binomial:
$$x^{-n}(1-x^{m-n})^{-1}$$
So far my expression is in the following form:
$$ \frac {m-nx^{m-n}}{1-x^{m-n}} $$
From here I think $m-n = 1$ and I've tried to prove using the fact that
$$\lim_{a \to \infty} x^a = 0, \space where \space \vert x \vert < 1$$ $$x^n > x^m, \space where \space m > n$$
I don't know if this is the correct way to prove that the expression equals $$\frac 1{1-x}$$ Thanks in advance and have a great day.