Trending proof for fairly simple fraction

Question

(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.

Welcome to the mystical world of Q Generalizations! (They are quite interesting to play with algebraically!) — Sidharth Ghoshal, Jul 27 '15 at 15:11
Note that $\forall x\in \mathbb R\forall n\in \mathbb N\left(x^n-1=(x-1)(x^{n-1}+x^{n-2}+\ldots +1)\right)$. — Git Gud, Jul 27 '15 at 15:12

lab bhattacharjee · Accepted Answer · 2015-07-27T15:14:04.507

2

$$\lim_{a\to1}\dfrac{a^b-1}{a-1}=\lim_{h\to0}\dfrac{(1+h)^b-1}h$$

Using Binomial series, this can be reduced to

$$\lim_{h\to0}\dfrac{1+bh+O(h^2)-1}h=?$$

edited Jul 27 '15 at 15:14

answered Jul 27 '15 at 15:11

lab bhattacharjee

274,582

This is even more efficient than L'Hopitals! – Sidharth Ghoshal Jul 27 '15 at 15:13
This calculates the derivative of $x\mapsto x^b$ at $x=1$. – Hagen von Eitzen Jul 27 '15 at 15:15

Trending proof for fairly simple fraction

1 Answers1