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log(x)/(x*log(10))
Limit of the function/ log(x)/(x*log(10))

Limit of the function log(x)/(x*log(10))

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You have entered [src] 
     /  log(x) \
 lim |---------|
x->oo\x*log(10)/
lim⁡x→∞(log⁡(x)xlog⁡(10))\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) 
Limit(log(x)/((x*log(10))), 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→∞(xlog⁡(10))=∞\lim_{x \to \infty}\left(x \log{\left(10 \right)}\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(log⁡(x)xlog⁡(10))\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(log⁡(x)xlog⁡(10))\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)
=
lim⁡x→∞(ddxlog⁡(x)ddxxlog⁡(10))\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} x \log{\left(10 \right)}}\right)
=
lim⁡x→∞(1xlog⁡(10))\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\right)
=
lim⁡x→∞(1xlog⁡(10))\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\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
02468-8-6-4-2-1010-2010
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Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(log⁡(x)xlog⁡(10))=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0
lim⁡x→0−(log⁡(x)xlog⁡(10))=∞\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = \infty
More at x→0 from the left
lim⁡x→0+(log⁡(x)xlog⁡(10))=−∞\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = -\infty
More at x→0 from the right
lim⁡x→1−(log⁡(x)xlog⁡(10))=0\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0
More at x→1 from the left
lim⁡x→1+(log⁡(x)xlog⁡(10))=0\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0
More at x→1 from the right
lim⁡x→−∞(log⁡(x)xlog⁡(10))=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0
More at x→-oo
The graph
Limit of the function log(x)/(x*log(10))
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