What is the limit as x approaches infinity of (pi/2-arctan(x))^{1/(ln(x))} ?

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limit as x approaches infinity of (pi/2-arctan(x))^{1/(ln(x))}

x \to \infty lim ((\frac{π}{2} - arctan (x))^{\frac{1}{l n ( x )}})

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

x \to \infty lim ((\frac{π}{2} - arctan (x))^{\frac{1}{l n ( x )}})

Apply exponent rules

= x \to \infty lim (e^{\frac{1}{l n ( x )} l n (\frac{π}{2} - a r c t a n (x))})

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

What is the limit as x approaches infinity of (pi/2-arctan(x))^{1/(ln(x))} ?
The limit as x approaches infinity of (pi/2-arctan(x))^{1/(ln(x))} is 1/e