What is the limit as x approaches 0-of 6(cot(x)-1/x) ?

The limit as x approaches 0-of 6(cot(x)-1/x) is 0

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

limit as x approaches 0-of 6(cot(x)-1/x)

x \to 0 - lim (6 (cot (x) - \frac{1}{x}))

0

Solution steps

x \to 0 - lim (6 (cot (x) - \frac{1}{x}))

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 6 \cdot x \to 0 - lim (cot (x) - \frac{1}{x})

Express with sin, cos

= 6 \cdot x \to 0 - lim (\frac{cos ( x )}{sin ( x )} - \frac{1}{x})

Simplify

\frac{cos ( x )}{sin ( x )} - \frac{1}{x} : \frac{x cos ( x ) - sin ( x )}{x sin ( x )}

= 6 \cdot x \to 0 - lim (\frac{x cos ( x ) - sin ( x )}{x sin ( x )})

Apply L'Hopital's Rule

: x \to 0 - lim (\frac{- x sin ( x )}{sin ( x ) + x cos ( x )})

= 6 \cdot x \to 0 - lim (\frac{- x sin ( x )}{sin ( x ) + x cos ( x )})

Apply L'Hopital's Rule

: x \to 0 - lim (\frac{- sin ( x ) - x cos ( x )}{2 cos ( x ) - x sin ( x )})

= 6 \cdot x \to 0 - lim (\frac{- sin ( x ) - x cos ( x )}{2 cos ( x ) - x sin ( x )})

Plug in the value

x = 0

= 6 \cdot \frac{- sin ( 0 ) - 0 \cdot cos ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

6 \cdot \frac{- sin ( 0 ) - 0 \cdot cos ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : 0

= 0

x \to 0 lim (2 (\frac{1}{x} - cot (x)))

x \to - 3 lim (\frac{1}{x ^{2}})

x \to 0 lim (π coth (\frac{1}{x}))

x \to 2 + lim (\frac{x ^{2} + 2}{x - 2})

t \to 0 lim (\frac{5 ^{t} - 1}{t})

What is the limit as x approaches 0-of 6(cot(x)-1/x) ?
The limit as x approaches 0-of 6(cot(x)-1/x) is 0