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log(1-x^3)/x
Limit of the function/ log(1-x^3)/x

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log(1-x^3)/x

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log(1 - x^3)/x
 
log(1 - x*3)/x
 
log(1 - x^3)/x
 
log(1 - x^3)/x

Limit of the function log(1-x^3)/x

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The solution

You have entered [src] 
     /   /     3\\
     |log\1 - x /|
 lim |-----------|
x->0+\     x     /
lim⁡x→0+(log⁡(−x3+1)x)\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) 
Limit(log(1 - x^3)/x, x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+log⁡(−x3+1)=0\lim_{x \to 0^+} \log{\left(- x^{3} + 1 \right)} = 0
and limit for the denominator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(log⁡(−x3+1)x)\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)
=
lim⁡x→0+(ddxlog⁡(−x3+1)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(- x^{3} + 1 \right)}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(−3x2−x3+1)\lim_{x \to 0^+}\left(- \frac{3 x^{2}}{- x^{3} + 1}\right)
=
lim⁡x→0+(−3x2)\lim_{x \to 0^+}\left(- 3 x^{2}\right)
=
lim⁡x→0+(−3x2)\lim_{x \to 0^+}\left(- 3 x^{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
02468-8-6-4-2-10102-2
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
One‐sided limits [src] 
     /   /     3\\
     |log\1 - x /|
 lim |-----------|
x->0+\     x     /
lim⁡x→0+(log⁡(−x3+1)x)\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) 
0
00 
= 8.98284195579066e-31
     /   /     3\\
     |log\1 - x /|
 lim |-----------|
x->0-\     x     /
lim⁡x→0−(log⁡(−x3+1)x)\lim_{x \to 0^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) 
0
00 
= 8.35837439466625e-30
= 8.35837439466625e-30
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(log⁡(−x3+1)x)=0\lim_{x \to 0^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0
More at x→0 from the left
lim⁡x→0+(log⁡(−x3+1)x)=0\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0
lim⁡x→∞(log⁡(−x3+1)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(log⁡(−x3+1)x)=−∞\lim_{x \to 1^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty
More at x→1 from the left
lim⁡x→1+(log⁡(−x3+1)x)=−∞\lim_{x \to 1^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty
More at x→1 from the right
lim⁡x→−∞(log⁡(−x3+1)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0
More at x→-oo
Numerical answer [src] 
8.98284195579066e-31
8.98284195579066e-31
The graph
Limit of the function log(1-x^3)/x
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