What is the limit as x approaches 1 of 1/((sec(\frac{pix){2})(1-sqrt(x)))} ?

The limit as x approaches 1 of 1/((sec(\frac{pix){2})(1-sqrt(x)))} is pi

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limit as x approaches 1 of 1/((sec(\frac{pix){2})(1-sqrt(x)))}

x \to 1 lim (\frac{1}{( sec ( \frac{π x}{2} ) ( 1 - x ) )})

π

Decimal

3.14159 \dots

Solution steps

x \to 1 lim (\frac{1}{sec ( \frac{π x}{2} ) ( 1 - x )})

Express with sin, cos

= x \to 1 lim (\frac{cos ( \frac{π x}{2} )}{1 - x})

Apply L'Hopital's Rule

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

Plug in the value

x = 1

= \frac{- sin ( \frac{π 1}{2} ) \frac{π}{2}}{- \frac{1}{2 1}}

Simplify

\frac{- sin ( \frac{π 1}{2} ) \frac{π}{2}}{- \frac{1}{2 1}} : π

= π

What is the limit as x approaches 1 of 1/((sec(\frac{pix){2})(1-sqrt(x)))} ?
The limit as x approaches 1 of 1/((sec(\frac{pix){2})(1-sqrt(x)))} is pi