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x^4*log(x)
Limit of the function/ x^4*log(x)

Limit of the function x^4*log(x)

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     / 4       \
 lim \x *log(x)/
x->0+
lim⁡x→0+(x4log⁡(x))\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right) 
Limit(x^4*log(x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+x4=0\lim_{x \to 0^+} x^{4} = 0
and limit for the denominator is
lim⁡x→0+1log⁡(x)=0\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(x4log⁡(x))\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right)
=
lim⁡x→0+(ddxx4ddx1log⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{4}}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}\right)
=
lim⁡x→0+(−4x4log⁡(x)2)\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\right)
=
lim⁡x→0+(−4x4log⁡(x)2)\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\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
02468-8-6-4-2-1010-2500025000
Plot the graph
One‐sided limits [src] 
     / 4       \
 lim \x *log(x)/
x->0+
lim⁡x→0+(x4log⁡(x))\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right) 
0
00 
= -4.17440384416252e-13
     / 4       \
 lim \x *log(x)/
x->0-
lim⁡x→0−(x4log⁡(x))\lim_{x \to 0^-}\left(x^{4} \log{\left(x \right)}\right) 
0
00 
= (-3.92017193388019e-13 + 2.09170932732045e-13j)
= (-3.92017193388019e-13 + 2.09170932732045e-13j)
Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(x4log⁡(x))=0\lim_{x \to 0^-}\left(x^{4} \log{\left(x \right)}\right) = 0
More at x→0 from the left
lim⁡x→0+(x4log⁡(x))=0\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right) = 0
lim⁡x→∞(x4log⁡(x))=∞\lim_{x \to \infty}\left(x^{4} \log{\left(x \right)}\right) = \infty
More at x→oo
lim⁡x→1−(x4log⁡(x))=0\lim_{x \to 1^-}\left(x^{4} \log{\left(x \right)}\right) = 0
More at x→1 from the left
lim⁡x→1+(x4log⁡(x))=0\lim_{x \to 1^+}\left(x^{4} \log{\left(x \right)}\right) = 0
More at x→1 from the right
lim⁡x→−∞(x4log⁡(x))=∞\lim_{x \to -\infty}\left(x^{4} \log{\left(x \right)}\right) = \infty
More at x→-oo
Numerical answer [src] 
-4.17440384416252e-13
-4.17440384416252e-13
The graph
Limit of the function x^4*log(x)
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