What is the limit as x approaches-infinity of (4arctan(8x))/(e^x+2) ?

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limit as x approaches-infinity of (4arctan(8x))/(e^x+2)

x \to - \infty lim (\frac{4 arctan ( 8 x )}{e ^{x} + 2})

- π

Decimal

- 3.14159 \dots

Solution steps

x \to - \infty lim (\frac{4 arctan ( 8 x )}{e ^{x} + 2})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 4 \cdot x \to - \infty lim (\frac{arctan ( 8 x )}{e ^{x} + 2})

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

= 4 \cdot \frac{lim _{x \to - \infty} ( arctan ( 8 x ) )}{lim _{x \to - \infty} ( e ^{x} + 2 )}

x \to - \infty lim (arctan (8 x)) = - \frac{π}{2}

x \to - \infty lim (e^{x} + 2) = 2

= 4 \cdot \frac{- \frac{π}{2}}{2}

Simplify

4 \cdot \frac{- \frac{π}{2}}{2} : - π

= - π

What is the limit as x approaches-infinity of (4arctan(8x))/(e^x+2) ?
The limit as x approaches-infinity of (4arctan(8x))/(e^x+2) is -pi