evaluate-lim-x-0-x-ln-sin-x-

Question Number 85127 by Rio Michael last updated on 19/Mar/20

evaluate :

\lim_{x \to 0} \sqrt{x} \ln (\sin x)

Commented by john santu last updated on 19/Mar/20

yes . i forgot \ln in this question

Commented by MJS last updated on 19/Mar/20

{there}^{'} s something wrong

Commented by jagoll last updated on 19/Mar/20

in which parts is wrong ?

Commented by MJS last updated on 19/Mar/20

(1) \lim_{x \to 0} \ln {(\sin x)}^{\sqrt{x}} \neq \lim_{x \to 0} {(1 + (\sin x - 1))}^{\sqrt{x}}

(2) how did you get the 3^{rd} line from the 2^{nd} one ?

\lim_{x \to 0} {({(1 + u)}^{\frac{1}{u}})}^{u v} \overset{?}{\Rightarrow} e^{\lim_{x \to 0} u v}

Answered by MJS last updated on 19/Mar/20

L = \lim_{x \to 0} \sqrt{x} \ln \sin x = \lim_{x \to 0} \frac{x \ln \sin x}{\sqrt{x}} =

= \lim_{x \to 0} \frac{\frac{d}{d x} [x \ln \sin x]}{\frac{d}{d x} [\sqrt{x}]} = \lim_{x \to 0} \frac{\frac{x}{\tan x} + \ln \sin x}{\frac{1}{2 \sqrt{x}}} =

= 2 \lim_{x \to 0} (\frac{\sqrt{x^{3}}}{\tan x} + \sqrt{x} \ln \sin x)

\Rightarrow

L = 2 L + 2 \lim_{x \to 0} \frac{\sqrt{x^{3}}}{\tan x}

\Rightarrow

L = - 2 \lim_{x \to 0} \frac{\sqrt{x^{3}}}{\tan x} = - 2 \lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x^{3}}]}{\frac{d}{d x} [\tan x]} =

= - 2 \lim_{x \to 0} \frac{\frac{3}{2} \sqrt{x}}{\frac{1}{\cos^{2} x}} = - 3 \lim_{x \to 0} \sqrt{x} \cos^{2} x = 0

Commented by Rio Michael last updated on 19/Mar/20

t h a n k s y^{'} a l l