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(-2+x)/(-8+x^3)
Limit of the function/ (-2+x)/(-8+x^3)

Limit of the function (-2+x)/(-8+x^3)

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     / -2 + x\
 lim |-------|
x->2+|      3|
     \-8 + x /
lim⁡x→2+(x−2x3−8)\lim_{x \to 2^+}\left(\frac{x - 2}{x^{3} - 8}\right) 
Limit((-2 + x)/(-8 + x^3), x, 2)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→2+(x−2)=0\lim_{x \to 2^+}\left(x - 2\right) = 0
and limit for the denominator is
lim⁡x→2+(x3−8)=0\lim_{x \to 2^+}\left(x^{3} - 8\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→2+(x−2x3−8)\lim_{x \to 2^+}\left(\frac{x - 2}{x^{3} - 8}\right)
=
lim⁡x→2+(ddx(x−2)ddx(x3−8))\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(x - 2\right)}{\frac{d}{d x} \left(x^{3} - 8\right)}\right)
=
lim⁡x→2+(13x2)\lim_{x \to 2^+}\left(\frac{1}{3 x^{2}}\right)
=
lim⁡x→2+112\lim_{x \to 2^+} \frac{1}{12}
=
lim⁡x→2+112\lim_{x \to 2^+} \frac{1}{12}
=
112\frac{1}{12}
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
-4.0-3.0-2.0-1.04.00.01.02.03.00.00.5
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→2−(x−2x3−8)=112\lim_{x \to 2^-}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{12}
More at x→2 from the left
lim⁡x→2+(x−2x3−8)=112\lim_{x \to 2^+}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{12}
lim⁡x→∞(x−2x3−8)=0\lim_{x \to \infty}\left(\frac{x - 2}{x^{3} - 8}\right) = 0
More at x→oo
lim⁡x→0−(x−2x3−8)=14\lim_{x \to 0^-}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{4}
More at x→0 from the left
lim⁡x→0+(x−2x3−8)=14\lim_{x \to 0^+}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{4}
More at x→0 from the right
lim⁡x→1−(x−2x3−8)=17\lim_{x \to 1^-}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{7}
More at x→1 from the left
lim⁡x→1+(x−2x3−8)=17\lim_{x \to 1^+}\left(\frac{x - 2}{x^{3} - 8}\right) = \frac{1}{7}
More at x→1 from the right
lim⁡x→−∞(x−2x3−8)=0\lim_{x \to -\infty}\left(\frac{x - 2}{x^{3} - 8}\right) = 0
More at x→-oo
Rapid solution [src] 
1/12
112\frac{1}{12}
Expand and simplify
One‐sided limits [src] 
     / -2 + x\
 lim |-------|
x->2+|      3|
     \-8 + x /
lim⁡x→2+(x−2x3−8)\lim_{x \to 2^+}\left(\frac{x - 2}{x^{3} - 8}\right) 
1/12
112\frac{1}{12} 
= 0.0833333333333333
     / -2 + x\
 lim |-------|
x->2-|      3|
     \-8 + x /
lim⁡x→2−(x−2x3−8)\lim_{x \to 2^-}\left(\frac{x - 2}{x^{3} - 8}\right) 
1/12
112\frac{1}{12} 
= 0.0833333333333333
= 0.0833333333333333
Numerical answer [src] 
0.0833333333333333
0.0833333333333333
The graph
Limit of the function (-2+x)/(-8+x^3)
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