calculate-lim-x-0-x-ln-1-e-sinx-

Question Number 41761 by math khazana by abdo last updated on 12/Aug/18

c a l c u l a t e {l i m}_{x \to 0} x l n (1 - e^{s i n x})

Answered by alex041103 last updated on 12/Aug/18

A s x \to 0 t h e e x p r e s s i o n b e c o m e s o n e

o f t h e f o r m 0 \times \infty w h i c h i s e a s i l u

t r a n s f o r m e d i n \frac{\infty}{\infty} :

L = \lim_{x \to 0} \frac{l n (1 - e^{s i n x})}{1 / x} = L^{'} H o p i t a l

= \lim_{x \to 0} \frac{\frac{- e^{s i n x} c o s x}{1 - e^{s i n x}}}{- 1 / x^{2}} = \lim_{x \to 0} \frac{x^{2}}{1 - e^{s i n (x)}} {c o s x e}^{s i n x} =

= \lim_{x \to 0} \frac{x^{2}}{1 - e^{s i n x}} 1

N o w w e a p p l y L^{'} H o p i t a l u n t i l w e f i n d

a r e s u l t :

L = \lim_{x \to 0} \frac{2 x}{- e^{s i n (x)} c o s (x)} = \frac{2 \times 0}{- e^{s i n (0)} c o s (0)} =

= \frac{0}{1 \times 1} = 0

\Rightarrow {l i m}_{x \to 0} x l n (1 - e^{s i n x}) = 0