What is the limit as x approaches 0 of (tan(x)((1-cos(x))))/(xxx) ?

The limit as x approaches 0 of (tan(x)((1-cos(x))))/(xxx) is 1/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

limit as x approaches 0 of (tan(x)((1-cos(x))))/(xxx)

x \to 0 lim (\frac{tan ( x ) ( ( 1 - cos ( x ) ) )}{xxx})

\frac{1}{2}

Decimal Notation

0.5

Solution steps

x \to 0 lim (\frac{tan ( x ) ( 1 - cos ( x ) )}{xxx})

Simplify

\frac{tan ( x ) ( 1 - cos ( x ) )}{xxx} : \frac{- sin ( x ) + tan ( x )}{x ^{3}}

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{sin ( x ) + 2 sec ^{2} ( x ) tan ( x )}{6 x})

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( x ) - 4 sec ^{2} ( x ) + 6 sec ^{4} ( x )}{6})

= x \to 0 lim (\frac{cos ( x ) - 4 sec ^{2} ( x ) + 6 sec ^{4} ( x )}{6})

Plug in the value

x = 0

= \frac{cos ( 0 ) - 4 sec ^{2} ( 0 ) + 6 sec ^{4} ( 0 )}{6}

Simplify

\frac{cos ( 0 ) - 4 sec ^{2} ( 0 ) + 6 sec ^{4} ( 0 )}{6} : \frac{1}{2}

= \frac{1}{2}

x \to 0 + lim (\frac{1}{tan ^{2} ( x )})

x \to 0 lim ((1 - 2 x)^{\frac{6}{x}})

x \to π lim (7 sin (x))

x \to \infty lim (e^{x} - 3 x)

x \to 1 + lim (\frac{3 x}{1 - x})

What is the limit as x approaches 0 of (tan(x)((1-cos(x))))/(xxx) ?
The limit as x approaches 0 of (tan(x)((1-cos(x))))/(xxx) is 1/2