What is the limit as x approaches 0 of (tan(4x))/(tan(9x)) ?

The limit as x approaches 0 of (tan(4x))/(tan(9x)) is 4/9

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limit as x approaches 0 of (tan(4x))/(tan(9x))

x \to 0 lim (\frac{tan ( 4 x )}{tan ( 9 x )})

\frac{4}{9}

Decimal

0.44444 \dots

Solution steps

x \to 0 lim (\frac{tan ( 4 x )}{tan ( 9 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{sec ^{2} ( 4 x ) \cdot 4}{sec ^{2} ( 9 x ) \cdot 9})

Plug in the value

x = 0

= \frac{sec ^{2} ( 4 \cdot 0 ) \cdot 4}{sec ^{2} ( 9 \cdot 0 ) \cdot 9}

Simplify

\frac{sec ^{2} ( 4 \cdot 0 ) \cdot 4}{sec ^{2} ( 9 \cdot 0 ) \cdot 9} : \frac{4}{9}

= \frac{4}{9}

What is the limit as x approaches 0 of (tan(4x))/(tan(9x)) ?
The limit as x approaches 0 of (tan(4x))/(tan(9x)) is 4/9