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

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

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limit as x approaches-infinity of 3^x*x

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

0

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

: x \to - \infty lim (\frac{1}{- ln ( 3 ) \cdot 3 ^{- x}})

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

Simplify

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

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

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

x \to - \infty lim (1) = 1

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

= - \frac{1}{ln ( 3 )} \cdot \frac{1}{\infty}

Simplify

- \frac{1}{ln ( 3 )} \cdot \frac{1}{\infty} : 0

= 0

x \to h lim (cot (x - h))

t \to 0 lim ((1 + t)^{\frac{1}{t}})

x \to 2 lim (\frac{2 - x}{4 - 2 x})

x \to 1 - lim (- 2)

x \to 4 lim (8)

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