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

The limit as x approaches 0 of x^2csc^2(2x) is 1/4

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

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

\frac{1}{4}

Decimal Notation

0.25

Solution steps

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

Express with sin, cos

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 x}{2 sin ( 4 x )})

= x \to 0 lim (\frac{2 x}{2 sin ( 4 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2}{8 cos ( 4 x )})

= x \to 0 lim (\frac{2}{8 cos ( 4 x )})

Plug in the value

x = 0

= \frac{2}{8 cos ( 4 \cdot 0 )}

Simplify

\frac{2}{8 cos ( 4 \cdot 0 )} : \frac{1}{4}

= \frac{1}{4}

\int 2 sin (2 x) - 2 cos (2 x) d x

e^{x^{2} + 1}, 0

\int 2 e^{- 1.5 x} d x

\frac{d}{d x} (\frac{e ^{x}}{x - 1})

\frac{d}{d x} (cos^{5} (3 x))

What is the limit as x approaches 0 of x^2csc^2(2x) ?
The limit as x approaches 0 of x^2csc^2(2x) is 1/4