What is the limit as x approaches 0 of x^2csc^2(3x) ?

The limit as x approaches 0 of x^2csc^2(3x) is 1/9

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limit as x approaches 0 of x^2csc^2(3x)

x \to 0 lim (x^{2} csc^{2} (3 x))

\frac{1}{9}

Decimal

0.11111 \dots

Solution steps

x \to 0 lim (x^{2} csc^{2} (3 x))

Express with sin, cos

= x \to 0 lim (\frac{x ^{2}}{sin ^{2} ( 3 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 x}{3 sin ( 6 x )})

= x \to 0 lim (\frac{2 x}{3 sin ( 6 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2}{18 cos ( 6 x )})

= x \to 0 lim (\frac{2}{18 cos ( 6 x )})

Plug in the value

x = 0

= \frac{2}{18 cos ( 6 \cdot 0 )}

Simplify

\frac{2}{18 cos ( 6 \cdot 0 )} : \frac{1}{9}

= \frac{1}{9}

z \to \infty lim (\frac{( a z + b ) ^{3}}{( cz + d ) ^{3}})

x \to 3 lim (4^{x})

x \to 2 lim (3 (1 - x) (2 - x))

x \to \infty lim (5 - \frac{1}{x})

x \to 0 lim (ln (x) cos (x))

What is the limit as x approaches 0 of x^2csc^2(3x) ?
The limit as x approaches 0 of x^2csc^2(3x) is 1/9