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

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

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

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

- 2

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

\frac{- 4 sec ^{2} ( 2 \cdot 0 )}{2} : - 2

= - 2

x^{2} y^{^{''}} + 9 x y^{^{'}} - 20 y = \frac{5}{x ^{3}}

f (x) = csc (x) (x + cot (x))

x \to 0 lim (\frac{1}{x + 5})

\frac{\partial}{\partial x} ((5 x - y)^{7})

\frac{d}{d x} (4 x^{\frac{5}{4}} + 6 x^{\frac{3}{2}} + 9 x)

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