lim-x-pi-e-sin-x-1-x-pi-

Question Number 80670 by jagoll last updated on 05/Feb/20

\lim_{x \to π} \frac{e^{\sin x} - 1}{x - π} = ?

Commented by jagoll last updated on 05/Feb/20

\lim_{x \to π} \frac{\cos x . e^{\sin x}}{1} = - 1

Commented by abdomathmax last updated on 05/Feb/20

l e t f (x) = \frac{e^{s i n x} - 1}{x - π} c h a n g e m e n t x - π = t g i v e

f (x) = g (t) = \frac{e^{s i n (π + t)} - 1}{t} = \frac{e^{- s i n t} - 1}{t}

\sim \frac{e^{- t} - 1}{t} \sim \frac{1 - t - 1}{t} = - 1 \Rightarrow {l i m}_{x \to π} f (x) = - 1

Answered by $@ty@m123 last updated on 05/Feb/20

L e t x - π = t

x = π + t

A s x \to π, t = 0

\lim_{t \to 0} \frac{e^{\sin (π + t)} - 1}{t}

\lim_{t \to 0} \frac{e^{- \sin t} - 1}{t}

\lim_{t \to 0} \frac{1 - \sin t + \frac{\sin^{2} t}{2!} + \dots . . - 1}{t}

\lim_{t \to 0} \frac{- \sin t}{t}

= - 1