lim-x-0-cos-x-cot-x-

Question Number 117637 by bemath last updated on 13/Oct/20

\lim_{x \to 0} {(\cos x)}^{\cot x} = ?

Answered by AbduraufKodiriy last updated on 13/Oct/20

\underset{x \to 0}{l i m} ({c o s x}^{c o t x}) = \underset{x \to 0}{l i m} ({c o s x}^{\frac{c o s x}{s i n x}}) =

= \underset{x \to 0}{l i m} {(1 + c o s x - 1)}^{\frac{c o s x s i n x}{1 - {c o s}^{2} x}} = \underset{x \to 0}{l i m} {({(1 + c o s x - 1)}^{\frac{1}{1 - c o s x}})}^{\frac{c o s x s i n x}{1 + c o s x}} =

= \underset{x \to 0}{l i m} (e^{\frac{c o s x s i n x}{1 + c o s x}}) = e^{0} = 1

Answered by Dwaipayan Shikari last updated on 13/Oct/20

\lim_{x \to 0} {(c o s x)}^{c o t x} = y

c o t x l o g (1 + c o s x - 1) = l o g y

c o t x (c o s x - 1) = l o g y \lim_{x \to 0} l o g (1 + x) = x

\frac{{c o s}^{2} x - c o s x}{s i n x} = l o g y

c o s x (\frac{2 {s i n}^{2} \frac{x}{2}}{2 s i n \frac{x}{2} c o s \frac{x}{2}}) = l o g y

l o g y = 0

y = e^{0} = 1