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

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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

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}})

Apply L'Hopital's Rule

= 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})

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}

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