What is the limit as x approaches 0 of (tan(3x)-sin(3x))/(x^3) ?

The limit as x approaches 0 of (tan(3x)-sin(3x))/(x^3) is 27/2

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limit as x approaches 0 of (tan(3x)-sin(3x))/(x^3)

x \to 0 lim (\frac{tan ( 3 x ) - sin ( 3 x )}{x ^{3}})

\frac{27}{2}

Decimal

13.5

Solution steps

x \to 0 lim (\frac{tan ( 3 x ) - sin ( 3 x )}{x ^{3}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{sec ^{2} ( 3 x ) \cdot 3 - cos ( 3 x ) \cdot 3}{3 x ^{2}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{18 sec ^{2} ( 3 x ) tan ( 3 x ) + 9 sin ( 3 x )}{6 x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{18 ( 6 sec ^{2} ( 3 x ) tan ^{2} ( 3 x ) + 3 sec ^{4} ( 3 x ) ) + 27 cos ( 3 x )}{6})

Plug in the value

x = 0

= \frac{18 ( 6 sec ^{2} ( 3 \cdot 0 ) tan ^{2} ( 3 \cdot 0 ) + 3 sec ^{4} ( 3 \cdot 0 ) ) + 27 cos ( 3 \cdot 0 )}{6}

Simplify

\frac{18 ( 6 sec ^{2} ( 3 \cdot 0 ) tan ^{2} ( 3 \cdot 0 ) + 3 sec ^{4} ( 3 \cdot 0 ) ) + 27 cos ( 3 \cdot 0 )}{6} : \frac{27}{2}

= \frac{27}{2}

What is the limit as x approaches 0 of (tan(3x)-sin(3x))/(x^3) ?
The limit as x approaches 0 of (tan(3x)-sin(3x))/(x^3) is 27/2