What is the limit as x approaches infinity of (e^{1/x})/(arctan(x)) ?

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

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limit as x approaches infinity of (e^{1/x})/(arctan(x))

x \to \infty lim (\frac{e ^{\frac{1}{x}}}{arctan ( x )})

\frac{2}{π}

Decimal Notation

0.63661 \dots

Solution steps

x \to \infty lim (\frac{e ^{\frac{1}{x}}}{arctan ( x )})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( e ^{\frac{1}{x}} )}{lim _{x \to \infty} ( arctan ( x ) )}

x \to \infty lim (e^{\frac{1}{x}}) = 1

x \to \infty lim (arctan (x)) = \frac{π}{2}

= \frac{1}{\frac{π}{2}}

Simplify

= \frac{2}{π}

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