What is the limit as x approaches 0 of ((3arccos(1/2+x)-3arccos(1/2)))/x ?

The limit as x approaches 0 of ((3arccos(1/2+x)-3arccos(1/2)))/x is -2sqrt(3)

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limit as x approaches 0 of ((3arccos(1/2+x)-3arccos(1/2)))/x

x \to 0 lim (\frac{( 3 arccos ( \frac{1}{2} + x ) - 3 arccos ( \frac{1}{2} ) )}{x})

- 23

Decimal

- 3.46410 \dots

Solution steps

x \to 0 lim (\frac{3 arccos ( \frac{1}{2} + x ) - 3 arccos ( \frac{1}{2} )}{x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- \frac{6}{- 4 x ^{2} - 4 x + 3}}{1})

Plug in the value

x = 0

= \frac{- \frac{6}{- 4 \cdot 0 ^{2} - 4 \cdot 0 + 3}}{1}

Simplify

\frac{- \frac{6}{- 4 \cdot 0 ^{2} - 4 \cdot 0 + 3}}{1} : - 23

= - 23

What is the limit as x approaches 0 of ((3arccos(1/2+x)-3arccos(1/2)))/x ?
The limit as x approaches 0 of ((3arccos(1/2+x)-3arccos(1/2)))/x is -2sqrt(3)