What is the limit as x approaches 0 of xcot((pix)/2) ?

The limit as x approaches 0 of xcot((pix)/2) is 2/pi

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limit as x approaches 0 of xcot((pix)/2)

x \to 0 lim (x cot (\frac{π x}{2}))

\frac{2}{π}

Decimal

0.63661 \dots

Solution steps

x \to 0 lim (x cot (\frac{π x}{2}))

Rewrite for L'Hopital

= x \to 0 lim \frac{x}{\frac{1}{c o t ( \frac{x π}{2} )}}

Apply L'Hopital's Rule

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

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( \frac{π 0}{2} ) \frac{π}{2}}

Simplify

\frac{1}{sec ^{2} ( \frac{π 0}{2} ) \frac{π}{2}} : \frac{2}{π}

= \frac{2}{π}

What is the limit as x approaches 0 of xcot((pix)/2) ?
The limit as x approaches 0 of xcot((pix)/2) is 2/pi