What is the limit as x approaches 0 of ((2arccos(x))/pi)^{1/x} ?

The limit as x approaches 0 of ((2arccos(x))/pi)^{1/x} is 1/(e^{2/pi)}

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limit as x approaches 0 of ((2arccos(x))/pi)^{1/x}

x \to 0 lim ((\frac{2 arccos ( x )}{π})^{\frac{1}{x}})

\frac{1}{e ^{\frac{2}{π}}}

Decimal

0.52907 \dots

Solution steps

x \to 0 lim ((\frac{2 arccos ( x )}{π})^{\frac{1}{x}})

Apply the following algebraic property

: a \cdot b = \frac{a}{\frac{1}{b}}, b \neq = 0

2 arccos (x) = \frac{arccos ( x )}{\frac{1}{2}}

= x \to 0 lim \frac{\frac{a r c c o s ( x )}{\frac{1}{2}}}{π}^{\frac{1}{x}}

Simplify

\frac{\frac{a r c c o s ( x )}{\frac{1}{2}}}{π}^{\frac{1}{x}} : (\frac{2}{π} arccos (x))^{\frac{1}{x}}

= x \to 0 lim ((\frac{2}{π} arccos (x))^{\frac{1}{x}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x} l n (\frac{2}{π} a r c c o s (x))})

Apply the Limit Chain Rule:

\frac{1}{e ^{\frac{2}{π}}}

= \frac{1}{e ^{\frac{2}{π}}}

What is the limit as x approaches 0 of ((2arccos(x))/pi)^{1/x} ?
The limit as x approaches 0 of ((2arccos(x))/pi)^{1/x} is 1/(e^{2/pi)}