What is the limit as x approaches-infinity of 3xe^x ?

The limit as x approaches-infinity of 3xe^x is 0

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limit as x approaches-infinity of 3xe^x

x \to - \infty lim (3 x e^{x})

0

Solution steps

x \to - \infty lim (3 x e^{x})

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

= 3 \cdot x \to - \infty lim (x e^{x})

Rewrite for L'Hopital

= 3 \cdot x \to - \infty lim (\frac{x}{\frac{1}{e ^{x}}})

Apply L'Hopital's Rule

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

= 3 \cdot x \to - \infty lim (\frac{1}{- e ^{- x}})

Simplify

\frac{1}{- e ^{- x}} : - e^{x}

= 3 \cdot x \to - \infty lim (- e^{x})

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

= 3 (- x \to - \infty lim (e^{x}))

Apply the common limit:

x \to - \infty lim (e^{x}) = 0

= 3 (- 0)

Simplify

= 0

x \to 2 lim (\frac{x + 1}{x + 2})

x \to 1 lim (arccos (\frac{1}{x + 1}))

x \to 0 + lim (\frac{x}{e ^{- \frac{1}{x}}})

x \to 2 lim (6 x^{2} + 3 x - 9)

x \to 0 lim (\frac{x ^{2}}{x ^{3}})

What is the limit as x approaches-infinity of 3xe^x ?
The limit as x approaches-infinity of 3xe^x is 0