What is the limit as x approaches 0 of (cos(2x)-cos(5x))/(x^2) ?

The limit as x approaches 0 of (cos(2x)-cos(5x))/(x^2) is 21/2

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limit as x approaches 0 of (cos(2x)-cos(5x))/(x^2)

x \to 0 lim (\frac{cos ( 2 x ) - cos ( 5 x )}{x ^{2}})

\frac{21}{2}

Decimal Notation

10.5

Solution steps

x \to 0 lim (\frac{cos ( 2 x ) - cos ( 5 x )}{x ^{2}})

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 4 cos ( 2 x ) + 25 cos ( 5 x )}{2})

= x \to 0 lim (\frac{- 4 cos ( 2 x ) + 25 cos ( 5 x )}{2})

Plug in the value

x = 0

= \frac{- 4 cos ( 2 \cdot 0 ) + 25 cos ( 5 \cdot 0 )}{2}

- 4 cos (2 \cdot 0) + 25 cos (5 \cdot 0) = 21

= \frac{21}{2}

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What is the limit as x approaches 0 of (cos(2x)-cos(5x))/(x^2) ?
The limit as x approaches 0 of (cos(2x)-cos(5x))/(x^2) is 21/2