lim-x-0-27-x-1-9-x-1-

Question Number 116037 by Rio Michael last updated on 30/Sep/20

\lim_{x \to 0} \frac{27^{x} - 1}{9^{x} - 1} = ? ?

Answered by bemath last updated on 30/Sep/20

l e t 3^{x} = t; t \to 1

\lim_{t \to 1} \frac{t^{3} - 1}{t^{2} - 1} = \lim_{t \to 1} \frac{(t - 1) (t^{2} + t + 1)}{(t - 1) (t + 1)} = \frac{3}{2}

Commented by Rio Michael last updated on 30/Sep/20

exactly .

\lim_{x \to 0} \frac{27^{x} - 1}{9^{x} - 1} = \lim_{x \to 0} \frac{(3^{x} - 1) (3^{2 x} + (3^{x}) + 1)}{(3^{x} - 1) (3^{x} + 1)} = \frac{3}{2}

Commented by bemath last updated on 30/Sep/20

y e s . .

Answered by Dwaipayan Shikari last updated on 30/Sep/20

\lim_{x \to 0} \frac{27^{x} - 1}{x} . \frac{x}{9^{x} - 1} = \frac{\log (27)}{\log (9)} = \frac{3}{2}