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exp(-x)*log(x)
Limit of the function/ exp(-x)*log(x)

Limit of the function exp(-x)*log(x)

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You have entered [src] 
     / -x       \
 lim \e  *log(x)/
x->oo
lim⁡x→∞(e−xlog⁡(x))\lim_{x \to \infty}\left(e^{- x} \log{\left(x \right)}\right) 
Limit(exp(-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→∞log⁡(x)=∞\lim_{x \to \infty} \log{\left(x \right)} = \infty
and limit for the denominator is
lim⁡x→∞ex=∞\lim_{x \to \infty} e^{x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(e−xlog⁡(x))\lim_{x \to \infty}\left(e^{- x} \log{\left(x \right)}\right)
=
lim⁡x→∞(ddxlog⁡(x)ddxex)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} e^{x}}\right)
=
lim⁡x→∞(e−xx)\lim_{x \to \infty}\left(\frac{e^{- x}}{x}\right)
=
lim⁡x→∞(e−xx)\lim_{x \to \infty}\left(\frac{e^{- x}}{x}\right)
=
00
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
02468-8-6-4-2-10102.5-2.5
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Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(e−xlog⁡(x))=0\lim_{x \to \infty}\left(e^{- x} \log{\left(x \right)}\right) = 0
lim⁡x→0−(e−xlog⁡(x))=−∞\lim_{x \to 0^-}\left(e^{- x} \log{\left(x \right)}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(e−xlog⁡(x))=−∞\lim_{x \to 0^+}\left(e^{- x} \log{\left(x \right)}\right) = -\infty
More at x→0 from the right
lim⁡x→1−(e−xlog⁡(x))=0\lim_{x \to 1^-}\left(e^{- x} \log{\left(x \right)}\right) = 0
More at x→1 from the left
lim⁡x→1+(e−xlog⁡(x))=0\lim_{x \to 1^+}\left(e^{- x} \log{\left(x \right)}\right) = 0
More at x→1 from the right
lim⁡x→−∞(e−xlog⁡(x))=∞\lim_{x \to -\infty}\left(e^{- x} \log{\left(x \right)}\right) = \infty
More at x→-oo
The graph
Limit of the function exp(-x)*log(x)
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