What is the limit as x approaches 0 of ((cos(5x)-1))/(xsin(x)) ?

The limit as x approaches 0 of ((cos(5x)-1))/(xsin(x)) is -25/2

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limit as x approaches 0 of ((cos(5x)-1))/(xsin(x))

x \to 0 lim (\frac{( cos ( 5 x ) - 1 )}{x sin ( x )})

- \frac{25}{2}

Decimal Notation

- 12.5

Solution steps

x \to 0 lim (\frac{cos ( 5 x ) - 1}{x sin ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 5 sin ( 5 x )}{sin ( x ) + x cos ( x )})

= x \to 0 lim (\frac{- 5 sin ( 5 x )}{sin ( x ) + x cos ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 25 cos ( 5 x )}{2 cos ( x ) - x sin ( x )})

= x \to 0 lim (\frac{- 25 cos ( 5 x )}{2 cos ( x ) - x sin ( x )})

Plug in the value

x = 0

= \frac{- 25 cos ( 5 \cdot 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

\frac{- 25 cos ( 5 \cdot 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : - \frac{25}{2}

= - \frac{25}{2}

x \to \infty lim ((\frac{x ^{3}}{x ^{3} + 4})^{x^{3}})

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

x \to 0 lim (\frac{sin ^{2} ( 9 x )}{x})

x \to 0 - lim (x^{2} ln (- x))

x \to 0 + lim (x^{3} e^{\frac{1}{x}})

What is the limit as x approaches 0 of ((cos(5x)-1))/(xsin(x)) ?
The limit as x approaches 0 of ((cos(5x)-1))/(xsin(x)) is -25/2