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 Notation

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}

x \to 0 + lim (x^{\frac{1}{f ( x )}})

x \to 0 - lim ((1 - e^{x})^{x})

x \to 0 lim ((1 - x^{2})^{\frac{1}{x ^{2}}})

x \to \infty lim (\frac{4 x ^{2} - 5 x}{1 - 3 x ^{2}})

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