What is the limit as x approaches pi/2 of ((cot^2(x)))/(1-sin(x)) ?

The limit as x approaches pi/2 of ((cot^2(x)))/(1-sin(x)) is 2

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limit as x approaches pi/2 of ((cot^2(x)))/(1-sin(x))

x \to \frac{π}{2} lim (\frac{( cot ^{2} ( x ) )}{1 - sin ( x )})

2

Solution steps

x \to \frac{π}{2} lim (\frac{cot ^{2} ( x )}{1 - sin ( x )})

Apply L'Hopital's Rule

: x \to \frac{π}{2} lim (\frac{- 2 csc ^{2} ( x ) cot ( x )}{- cos ( x )})

= x \to \frac{π}{2} lim (\frac{- 2 csc ^{2} ( x ) cot ( x )}{- cos ( x )})

Apply L'Hopital's Rule

: x \to \frac{π}{2} lim (\frac{- 2 ( - 2 csc ^{2} ( x ) cot ^{2} ( x ) - csc ^{4} ( x ) )}{sin ( x )})

= x \to \frac{π}{2} lim (\frac{- 2 ( - 2 csc ^{2} ( x ) cot ^{2} ( x ) - csc ^{4} ( x ) )}{sin ( x )})

Plug in the value

x = \frac{π}{2}

= \frac{- 2 ( - 2 csc ^{2} ( \frac{π}{2} ) cot ^{2} ( \frac{π}{2} ) - csc ^{4} ( \frac{π}{2} ) )}{sin ( \frac{π}{2} )}

Simplify

\frac{- 2 ( - 2 csc ^{2} ( \frac{π}{2} ) cot ^{2} ( \frac{π}{2} ) - csc ^{4} ( \frac{π}{2} ) )}{sin ( \frac{π}{2} )} : 2

= 2

x \to c lim (e^{x})

x \to + 0 lim (coth (x))

x \to 1 lim (4 x - a x + b - 2)

x \to 8 lim (\frac{1}{x + 8})

x \to 0 + lim (x ln (sin (2 x)))

What is the limit as x approaches pi/2 of ((cot^2(x)))/(1-sin(x)) ?
The limit as x approaches pi/2 of ((cot^2(x)))/(1-sin(x)) is 2