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Limit of the function 3^(-x)*log(x)

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     / -x       \
 lim \3  *log(x)/
x->oo

\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right)

Limit(3^(-x)*log(x), x, oo, dir='-')

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

\lim_{x \to \infty} \log{\left(x \right)} = \infty

and limit for the denominator is

\lim_{x \to \infty} 3^{x} = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} 3^{x}}\right)

\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)

\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)

0

It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)

The graph

Plot the graph

Rapid solution [src]

0

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right) = 0

\lim_{x \to 0^-}\left(3^{- x} \log{\left(x \right)}\right) = -\infty

\lim_{x \to 0^+}\left(3^{- x} \log{\left(x \right)}\right) = -\infty

\lim_{x \to 1^-}\left(3^{- x} \log{\left(x \right)}\right) = 0

\lim_{x \to 1^+}\left(3^{- x} \log{\left(x \right)}\right) = 0

\lim_{x \to -\infty}\left(3^{- x} \log{\left(x \right)}\right) = \infty

The graph