What is the limit as x approaches 0 of (sin(8x))/(tan(9x)) ?

The limit as x approaches 0 of (sin(8x))/(tan(9x)) is 8/9

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limit as x approaches 0 of (sin(8x))/(tan(9x))

x \to 0 lim (\frac{sin ( 8 x )}{tan ( 9 x )})

\frac{8}{9}

Decimal

0.88888 \dots

Solution steps

x \to 0 lim (\frac{sin ( 8 x )}{tan ( 9 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cos ( 8 x ) \cdot 8}{sec ^{2} ( 9 x ) \cdot 9})

Plug in the value

x = 0

= \frac{cos ( 8 \cdot 0 ) \cdot 8}{sec ^{2} ( 9 \cdot 0 ) \cdot 9}

Simplify

\frac{cos ( 8 \cdot 0 ) \cdot 8}{sec ^{2} ( 9 \cdot 0 ) \cdot 9} : \frac{8}{9}

= \frac{8}{9}

What is the limit as x approaches 0 of (sin(8x))/(tan(9x)) ?
The limit as x approaches 0 of (sin(8x))/(tan(9x)) is 8/9