What is the limit as x approaches pi/2 of (pi/(cos(x)))-2xtan(x) ?

The limit as x approaches pi/2 of (pi/(cos(x)))-2xtan(x) is 2

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limit as x approaches pi/2 of (pi/(cos(x)))-2xtan(x)

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

2

Solution steps

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

Express with sin, cos

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

Apply L'Hopital's Rule

: x \to \frac{π}{2} lim (\frac{- 2 ( x cos ( x ) + sin ( x ) )}{- sin ( x )})

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

Plug in the value

x = \frac{π}{2}

= \frac{- 2 ( \frac{π}{2} cos ( \frac{π}{2} ) + sin ( \frac{π}{2} ) )}{- sin ( \frac{π}{2} )}

Simplify

\frac{- 2 ( \frac{π}{2} cos ( \frac{π}{2} ) + sin ( \frac{π}{2} ) )}{- sin ( \frac{π}{2} )} : 2

= 2

x \to \infty lim (\frac{5 ^{x + 1}}{e ^{x}})

x \to \infty lim (6 x e^{- x})

x \to 2 lim (\frac{2 x - 1}{3 x - 5})

x \to \infty lim (7 x^{\frac{l n ( 2 )}{7 + l n ( x )}})

x \to 0 - lim (\frac{sin ( 3 x )}{x})

What is the limit as x approaches pi/2 of (pi/(cos(x)))-2xtan(x) ?
The limit as x approaches pi/2 of (pi/(cos(x)))-2xtan(x) is 2