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Limit of the function 2^(-x)log(2x)

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     / -x         \
 lim \2  *log(2*x)/
x->oo

\lim_{x \to \infty}\left(2^{- x} \log{\left(2 x \right)}\right)

Limit(log(2*x)/2^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(2 x \right)} = \infty

and limit for the denominator is

\lim_{x \to \infty} 2^{x} = \infty

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

\lim_{x \to \infty}\left(2^{- x} \log{\left(2 x \right)}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(2 x \right)}}{\frac{d}{d x} 2^{x}}\right)

\lim_{x \to \infty}\left(\frac{2^{- x}}{x \log{\left(2 \right)}}\right)

\lim_{x \to \infty}\left(\frac{2^{- x}}{x \log{\left(2 \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(2^{- x} \log{\left(2 x \right)}\right) = 0

\lim_{x \to 0^-}\left(2^{- x} \log{\left(2 x \right)}\right) = -\infty

\lim_{x \to 0^+}\left(2^{- x} \log{\left(2 x \right)}\right) = -\infty

\lim_{x \to 1^-}\left(2^{- x} \log{\left(2 x \right)}\right) = \frac{\log{\left(2 \right)}}{2}

\lim_{x \to 1^+}\left(2^{- x} \log{\left(2 x \right)}\right) = \frac{\log{\left(2 \right)}}{2}

\lim_{x \to -\infty}\left(2^{- x} \log{\left(2 x \right)}\right) = \infty

The graph

Limit of the function 2^(-x)*log(2*x)