What is the limit as x approaches 0 of (tan(x))/(x^2cot(3x)) ?

The limit as x approaches 0 of (tan(x))/(x^2cot(3x)) is 3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

limit as x approaches 0 of (tan(x))/(x^2cot(3x))

x \to 0 lim (\frac{tan ( x )}{x ^{2} cot ( 3 x )})

3

Solution steps

x \to 0 lim (\frac{tan ( x )}{x ^{2} cot ( 3 x )})

Express with sin, cos

= x \to 0 lim (\frac{sin ( x ) sin ( 3 x )}{x ^{2} cos ( 3 x ) cos ( x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cos ( x ) sin ( 3 x ) + cos ( 3 x ) \cdot 3 sin ( x )}{2 x cos ( 3 x ) cos ( x ) + ( - 3 sin ( 3 x ) cos ( x ) - sin ( x ) cos ( 3 x ) ) x ^{2}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- 10 sin ( x ) sin ( 3 x ) + 6 cos ( x ) cos ( 3 x )}{- 10 x ^{2} cos ( x ) cos ( 3 x ) + 6 x ^{2} sin ( x ) sin ( 3 x ) - 12 x sin ( 3 x ) cos ( x ) - 4 x sin ( x ) cos ( 3 x ) + 2 cos ( 3 x ) cos ( x )})

Plug in the value

x = 0

= \frac{- 10 sin ( 0 ) sin ( 3 \cdot 0 ) + 6 cos ( 0 ) cos ( 3 \cdot 0 )}{- 10 \cdot 0 ^{2} cos ( 0 ) cos ( 3 \cdot 0 ) + 6 \cdot 0 ^{2} sin ( 0 ) sin ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) cos ( 0 ) - 4 \cdot 0 \cdot sin ( 0 ) cos ( 3 \cdot 0 ) + 2 cos ( 3 \cdot 0 ) cos ( 0 )}

Simplify

\frac{- 10 sin ( 0 ) sin ( 3 \cdot 0 ) + 6 cos ( 0 ) cos ( 3 \cdot 0 )}{- 10 \cdot 0 ^{2} cos ( 0 ) cos ( 3 \cdot 0 ) + 6 \cdot 0 ^{2} sin ( 0 ) sin ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) cos ( 0 ) - 4 \cdot 0 \cdot sin ( 0 ) cos ( 3 \cdot 0 ) + 2 cos ( 3 \cdot 0 ) cos ( 0 )} : 3

= 3

What is the limit as x approaches 0 of (tan(x))/(x^2cot(3x)) ?
The limit as x approaches 0 of (tan(x))/(x^2cot(3x)) is 3