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

The limit as x approaches 0 of ((x^3))/(tan^3(2x)) is 1/8

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limit as x approaches 0 of ((x^3))/(tan^3(2x))

x \to 0 lim (\frac{( x ^{3} )}{tan ^{3} ( 2 x )})

\frac{1}{8}

Decimal Notation

0.125

Solution steps

x \to 0 lim (\frac{x ^{3}}{tan ^{3} ( 2 x )})

Simplify

\frac{x ^{3}}{tan ^{3} ( 2 x )} : (\frac{x}{tan ( 2 x )})^{3}

= x \to 0 lim ((\frac{x}{tan ( 2 x )})^{3})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= (x \to 0 lim (\frac{x}{tan ( 2 x )}))^{3}

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{sec ^{2} ( 2 x ) \cdot 2})

= (x \to 0 lim (\frac{1}{sec ^{2} ( 2 x ) \cdot 2}))^{3}

Plug in the value

x = 0

= (\frac{1}{sec ^{2} ( 2 \cdot 0 ) \cdot 2})^{3}

Simplify

(\frac{1}{sec ^{2} ( 2 \cdot 0 ) \cdot 2})^{3} : \frac{1}{8}

= \frac{1}{8}

\int_{1}^{5} \frac{1}{x ^{2}} d x

\int (3 x^{2} + 1) d x

\frac{1}{( x - 1 )}

\int (x^{2} + 2 x) e^{x^{3} + 3 x^{2} - 1} d x

f (x) = (x - 1)^{2} + 1, at x = - 1

What is the limit as x approaches 0 of ((x^3))/(tan^3(2x)) ?
The limit as x approaches 0 of ((x^3))/(tan^3(2x)) is 1/8