What is the limit as x approaches 0 of xcsc(6x) ?

The limit as x approaches 0 of xcsc(6x) is 1/6

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limit as x approaches 0 of xcsc(6x)

x \to 0 lim (x csc (6 x))

\frac{1}{6}

Decimal

0.16666 \dots

Solution steps

x \to 0 lim (x csc (6 x))

Express with sin, cos

= x \to 0 lim (\frac{x}{sin ( 6 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 6 x ) \cdot 6})

= x \to 0 lim (\frac{1}{cos ( 6 x ) \cdot 6})

Plug in the value

x = 0

= \frac{1}{cos ( 6 \cdot 0 ) \cdot 6}

cos (6 \cdot 0) \cdot 6 = 6

= \frac{1}{6}

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

x \to \infty lim (\frac{2 x}{3})

x \to 11 lim (- \frac{10}{x - 11})

x \to 0 lim (\frac{csc ( - 6 x )}{csc ( 10 x )})

x \to 0 lim (\frac{ln ( 1 + 6 x )}{x})

What is the limit as x approaches 0 of xcsc(6x) ?
The limit as x approaches 0 of xcsc(6x) is 1/6