find-lim-x-0-ln-sin-xcos-1-x-1-if-it-exits-

Question Number 85057 by kushdasbaghar@gmail.com last updated on 18/Mar/20

f i n d \underset{x \to 0}{\lim l n} (\sin x \cos \frac{1}{x} + 1) i f i t e x i t s .

Commented by abdomathmax last updated on 19/Mar/20

w e h a v e l n (1 + s i n x c o s (\frac{1}{x})) \sim s i n x c o s (\frac{1}{x}) (x \to 0)

s i n x c o s (\frac{1}{x}) \sim x c o s (\frac{1}{x}) a n d ∣ x c o s (\frac{1}{x}) ∣⩽∣ x ∣\to 0 \Rightarrow

{l i m}_{x \to 0} s i n x c o s (\frac{1}{x}) = 0 \Rightarrow {l i m}_{x \to 0} l n (s i n x c o s (\frac{1}{x}) + 1) = 0