What is the limit as x approaches pi/2 of (csc(4x))/(csc(2x)) ?

The limit as x approaches pi/2 of (csc(4x))/(csc(2x)) is -1/2

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limit as x approaches pi/2 of (csc(4x))/(csc(2x))

x \to \frac{π}{2} lim (\frac{csc ( 4 x )}{csc ( 2 x )})

- \frac{1}{2}

Decimal Notation

- 0.5

Solution steps

x \to \frac{π}{2} lim (\frac{csc ( 4 x )}{csc ( 2 x )})

Express with sin, cos

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

Apply L'Hopital's Rule

: x \to \frac{π}{2} lim (\frac{cos ( 2 x ) \cdot 2}{cos ( 4 x ) \cdot 4})

= x \to \frac{π}{2} lim (\frac{cos ( 2 x ) \cdot 2}{cos ( 4 x ) \cdot 4})

Plug in the value

x = \frac{π}{2}

= \frac{cos ( 2 \cdot \frac{π}{2} ) \cdot 2}{cos ( 4 \cdot \frac{π}{2} ) \cdot 4}

Simplify

\frac{cos ( 2 \cdot \frac{π}{2} ) \cdot 2}{cos ( 4 \cdot \frac{π}{2} ) \cdot 4} : - \frac{1}{2}

= - \frac{1}{2}

f (x) = \frac{2 x - 1}{x + 5}, at x = 1

x \to - \infty lim (\frac{4 x ^{6}}{x ^{3} - 8})

θ \to \frac{π}{3} lim (\frac{tan ( θ )}{sin ^{2} ( θ )})

\int \frac{x ^{2}}{( x + 1 ) ^{50}} d x

\frac{\partial}{\partial x} (2 x e^{- y})

What is the limit as x approaches pi/2 of (csc(4x))/(csc(2x)) ?
The limit as x approaches pi/2 of (csc(4x))/(csc(2x)) is -1/2