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

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

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

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

4

Solution steps

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

Express with sin, cos

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

Simplify

\frac{x ^{2}}{sin ^{2} ( \frac{x}{2} )} : (\frac{x}{sin ( \frac{x}{2} )})^{2}

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

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( \frac{x}{2} ) \frac{1}{2}})

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

Plug in the value

x = 0

= (\frac{1}{cos ( \frac{0}{2} ) \frac{1}{2}})^{2}

Simplify

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

= 4

(- 9, 6), (- 6, - 9)

\int r^{4} (7 - \frac{r ^{5}}{10})^{3} d r

y^{^{'}} = \frac{3 t ^{2} - e ^{t}}{2 y - 5}

\int 3 t (5 t^{2} - 3 t + 2) d t

n = 1 \sum \infty 3 n + 1

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