What is the limit as x approaches 0 of 2x*cot(3x) ?

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

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limit as x approaches 0 of 2x*cot(3x)

x \to 0 lim (2 x \cdot cot (3 x))

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

x \to 0 lim (2 x cot (3 x))

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

= 2 \cdot x \to 0 lim (x cot (3 x))

Rewrite for L'Hopital

= 2 \cdot x \to 0 lim (\frac{x}{\frac{1}{c o t ( 3 x )}})

Apply L'Hopital's Rule

= 2 \cdot x \to 0 lim (\frac{1}{sec ^{2} ( 3 x ) \cdot 3})

Plug in the value

x = 0

= 2 \cdot \frac{1}{sec ^{2} ( 3 \cdot 0 ) \cdot 3}

Simplify

2 \cdot \frac{1}{sec ^{2} ( 3 \cdot 0 ) \cdot 3} : \frac{2}{3}

= \frac{2}{3}

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