What is the limit as x approaches 0 of (ln(cos(3x)))/(ln(cos(5x))) ?

The limit as x approaches 0 of (ln(cos(3x)))/(ln(cos(5x))) is 9/25

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limit as x approaches 0 of (ln(cos(3x)))/(ln(cos(5x)))

x \to 0 lim (\frac{ln ( cos ( 3 x ) )}{ln ( cos ( 5 x ) )})

\frac{9}{25}

Decimal

0.36

Solution steps

x \to 0 lim (\frac{ln ( cos ( 3 x ) )}{ln ( cos ( 5 x ) )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- 3 tan ( 3 x )}{- 5 tan ( 5 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- 9 sec ^{2} ( 3 x )}{- 25 sec ^{2} ( 5 x )})

Plug in the value

x = 0

= \frac{- 9 sec ^{2} ( 3 \cdot 0 )}{- 25 sec ^{2} ( 5 \cdot 0 )}

Simplify

\frac{- 9 sec ^{2} ( 3 \cdot 0 )}{- 25 sec ^{2} ( 5 \cdot 0 )} : \frac{9}{25}

= \frac{9}{25}

What is the limit as x approaches 0 of (ln(cos(3x)))/(ln(cos(5x))) ?
The limit as x approaches 0 of (ln(cos(3x)))/(ln(cos(5x))) is 9/25