lim-x-2-5-x-25-x-2-with-out-H-L-roule-

Question Number 185508 by mathlove last updated on 23/Jan/23

\lim_{x \to 2} \frac{5^{x} - 25}{x - 2} = ?

w i t h o u t H^{'} L r o u l e

Answered by Ar Brandon last updated on 23/Jan/23

\lim_{x \to 2} \frac{5^{x} - 25}{x - 2} = \lim_{t \to 0} \frac{5^{t + 2} - 25}{t} = \lim_{t \to 0} \frac{25 (5^{t} - 1)}{t}

= 25 \lim_{t \to 0} \frac{e^{t \ln 5} - 1}{t} = 25 \lim_{t \to 0} \frac{(1 + t \ln 5) - 1}{t}

= 25 \lim_{t \to 0} \frac{t \ln 5}{t} = 25 \lim_{t \to 0} (\ln 5) = 25 \ln (5)