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

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

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

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

1

Solution steps

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

Rewrite for L'Hopital

= x \to 0 + lim (\frac{x ^{2}}{\frac{1}{c s c ^{2} ( x )}})

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{2}{cos ( 2 x ) \cdot 2})

= x \to 0 + lim (\frac{2}{cos ( 2 x ) \cdot 2})

Plug in the value

x = 0

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

Simplify

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

= 1

x \frac{d y}{d x} = y - \frac{1}{y}

\frac{d}{d x} (\frac{x ^{4} - 5 x ^{3} + x}{x ^{2}})

\frac{d y}{d x} = \frac{e ^{x} + 1}{2 y}

x \to - \infty lim (4 x)

\int \frac{1}{3 - 2 x} d x

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