What is the limit as x approaches 0 of sin(3x)cot(x) ?

The limit as x approaches 0 of sin(3x)cot(x) is 3

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limit as x approaches 0 of sin(3x)cot(x)

x \to 0 lim (sin (3 x) cot (x))

3

Solution steps

x \to 0 lim (sin (3 x) cot (x))

Express with sin, cos

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

= \frac{- sin ( 0 ) sin ( 3 \cdot 0 ) + 3 cos ( 3 \cdot 0 ) cos ( 0 )}{cos ( 0 )}

Simplify

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

= 3

What is the limit as x approaches 0 of sin(3x)cot(x) ?
The limit as x approaches 0 of sin(3x)cot(x) is 3