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

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

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

i.e. limit for the numerator is
lim⁡x→1+(x−1)=0\lim_{x \to 1^+}\left(x - 1\right) = 0
and limit for the denominator is
lim⁡x→1+(x3−1)=0\lim_{x \to 1^+}\left(x^{3} - 1\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(x−1x3−1)\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right)
=
lim⁡x→1+(ddx(x−1)ddx(x3−1))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \left(x^{3} - 1\right)}\right)
=
lim⁡x→1+(13x2)\lim_{x \to 1^+}\left(\frac{1}{3 x^{2}}\right)
=
lim⁡x→1+13\lim_{x \to 1^+} \frac{1}{3}
=
lim⁡x→1+13\lim_{x \to 1^+} \frac{1}{3}
=
13\frac{1}{3}
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
-2.0-1.5-1.0-0.52.00.00.51.01.502
Plot the graph
Rapid solution [src] 
1/3
13\frac{1}{3}
Expand and simplify
One‐sided limits [src] 
     / -1 + x\
 lim |-------|
x->1+|      3|
     \-1 + x /
lim⁡x→1+(x−1x3−1)\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right) 
1/3
13\frac{1}{3} 
= 0.333333333333333
     / -1 + x\
 lim |-------|
x->1-|      3|
     \-1 + x /
lim⁡x→1−(x−1x3−1)\lim_{x \to 1^-}\left(\frac{x - 1}{x^{3} - 1}\right) 
1/3
13\frac{1}{3} 
= 0.333333333333333
= 0.333333333333333
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(x−1x3−1)=13\lim_{x \to 1^-}\left(\frac{x - 1}{x^{3} - 1}\right) = \frac{1}{3}
More at x→1 from the left
lim⁡x→1+(x−1x3−1)=13\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right) = \frac{1}{3}
lim⁡x→∞(x−1x3−1)=0\lim_{x \to \infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0
More at x→oo
lim⁡x→0−(x−1x3−1)=1\lim_{x \to 0^-}\left(\frac{x - 1}{x^{3} - 1}\right) = 1
More at x→0 from the left
lim⁡x→0+(x−1x3−1)=1\lim_{x \to 0^+}\left(\frac{x - 1}{x^{3} - 1}\right) = 1
More at x→0 from the right
lim⁡x→−∞(x−1x3−1)=0\lim_{x \to -\infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0
More at x→-oo
Numerical answer [src] 
0.333333333333333
0.333333333333333
The graph
Limit of the function (-1+x)/(-1+x^3)
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