What is the limit as x approaches 0 of (1-2cos(x)+cos(2x))/(x^2) ?

The limit as x approaches 0 of (1-2cos(x)+cos(2x))/(x^2) is -1

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limit as x approaches 0 of (1-2cos(x)+cos(2x))/(x^2)

x \to 0 lim (\frac{1 - 2 cos ( x ) + cos ( 2 x )}{x ^{2}})

- 1

Solution steps

x \to 0 lim (\frac{1 - 2 cos ( x ) + cos ( 2 x )}{x ^{2}})

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

\frac{2 cos ( 0 ) - 4 cos ( 2 \cdot 0 )}{2} : - 1

= - 1

y = - \frac{2}{5} x - 3

\frac{d}{d x} (e^{c o s (2 x)} sin (2 x))

\int - 6 x^{2} d x

x \to \infty lim (\frac{2 e ^{2 x}}{2 x})

x \to 0 lim (cos (\frac{9}{x}))

What is the limit as x approaches 0 of (1-2cos(x)+cos(2x))/(x^2) ?
The limit as x approaches 0 of (1-2cos(x)+cos(2x))/(x^2) is -1