What is the limit as x approaches 0 of (2sin(x)-sin(2x))/(sin(x)-xcos(x)) ?

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 (2sin(x)-sin(2x))/(sin(x)-xcos(x))

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

3

Solution steps

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- 2 sin ( x ) + 4 sin ( 2 x )}{sin ( x ) + x cos ( x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- 2 cos ( x ) + 8 cos ( 2 x )}{2 cos ( x ) - x sin ( x )})

Plug in the value

x = 0

= \frac{- 2 cos ( 0 ) + 8 cos ( 2 \cdot 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

\frac{- 2 cos ( 0 ) + 8 cos ( 2 \cdot 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : 3

= 3

What is the limit as x approaches 0 of (2sin(x)-sin(2x))/(sin(x)-xcos(x)) ?
The limit as x approaches 0 of (2sin(x)-sin(2x))/(sin(x)-xcos(x)) is 3