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

The limit as x approaches 0 of (ln(cos(x)))/(ln(cos(2x))) is 1/4

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

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

\frac{1}{4}

Decimal

0.25

Solution steps

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= \frac{1}{4}

What is the limit as x approaches 0 of (ln(cos(x)))/(ln(cos(2x))) ?
The limit as x approaches 0 of (ln(cos(x)))/(ln(cos(2x))) is 1/4