(Please humour the physicist!)
Why does $\left(\frac{1-a}{1-a^b}\right) \to \frac{1}{b}$ as $a \to 1$?
This came from a calculation involving flow measurement of gases, and although I can see and compute the answer, I don't understand how this general result occurs.
$$\lim_{a\to1}\dfrac{a^b-1}{a-1}=\lim_{h\to0}\dfrac{(1+h)^b-1}h$$
Using Binomial series, this can be reduced to
$$\lim_{h\to0}\dfrac{1+bh+O(h^2)-1}h=?$$