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

The limit as x approaches 0 of xcsc(3x) is 1/3

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limit as x approaches 0 of xcsc(3x)

x \to 0 lim (x csc (3 x))

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

x \to 0 lim (x csc (3 x))

Express with sin, cos

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 3 x ) \cdot 3})

= x \to 0 lim (\frac{1}{cos ( 3 x ) \cdot 3})

Plug in the value

x = 0

= \frac{1}{cos ( 3 \cdot 0 ) \cdot 3}

cos (3 \cdot 0) \cdot 3 = 3

= \frac{1}{3}

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What is the limit as x approaches 0 of xcsc(3x) ?
The limit as x approaches 0 of xcsc(3x) is 1/3