What is the limit as x approaches 2 of (x^2-4}{xcos(\frac{xpi)/2)-2cos((xpi)/2)} ?

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limit as x approaches 2 of (x^2-4}{xcos(\frac{xpi)/2)-2cos((xpi)/2)}

x \to 2 lim (\frac{x ^{2} - 4}{x cos ( \frac{x π}{2} ) - 2 cos ( \frac{x π}{2} )})

- 4

Solution steps

x \to 2 lim (\frac{x ^{2} - 4}{x cos ( \frac{x π}{2} ) - 2 cos ( \frac{x π}{2} )})

Use the following identity:

cos (x) = 1 - 2 sin^{2} (\frac{x}{2})

= x \to 2 lim \frac{x ^{2} - 4}{x ( 1 - 2 sin ^{2} ( \frac{\frac{x π}{2}}{2} ) ) - 2 ( 1 - 2 sin ^{2} ( \frac{\frac{x π}{2}}{2} ) )}

Simplify

\frac{x ^{2} - 4}{x ( 1 - 2 sin ^{2} ( \frac{\frac{x π}{2}}{2} ) ) - 2 ( 1 - 2 sin ^{2} ( \frac{\frac{x π}{2}}{2} ) )} : sec (\frac{π x}{2}) (2 + x)

= x \to 2 lim (sec (\frac{π x}{2}) (2 + x))

Plug in the value

x = 2

= sec (\frac{π 2}{2}) (2 + 2)

Simplify

sec (\frac{π 2}{2}) (2 + 2) : - 4

= - 4

- 2 x^{2} y + y^{^{'}} = - 7 x^{2}

x \cdot (3 \cdot x - 4)^{0.5}

\frac{\partial}{\partial y} (x z - 5 x^{2} y^{3} z^{4})

\frac{d}{d x} (\frac{- 10 x + 4}{3 x - 8})

y = x^{2} + x

What is the limit as x approaches 2 of (x^2-4}{xcos(\frac{xpi)/2)-2cos((xpi)/2)} ?
The limit as x approaches 2 of (x^2-4}{xcos(\frac{xpi)/2)-2cos((xpi)/2)} is -4