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

The limit as x approaches 1 of (cos(pi/2 x))/(x-1) is -pi/2

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limit as x approaches 1 of (cos(pi/2 x))/(x-1)

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

- \frac{π}{2}

Decimal Notation

- 1.57079 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 1

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

Simplify

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

= - \frac{π}{2}

x \to 0 lim ((1 + \frac{x}{4})^{\frac{1}{x}})

x \to - 1 lim (x^{3} - 3 x + 2)

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

x \to 0 lim (\frac{1}{arcsin ( x )})

x \to 100 lim ((2 x)^{0})

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