Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (5-4*x+3*x^2)/(1-x+2*x^2)
-
Limit of (1-7/x)^x
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Limit of ((1+x^2)/(-1+x^2))^(x^2)
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Limit of (-1+x)/(-1+x^3)
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Identical expressions
- (- one +x)/(- one +x^ three)
- ( minus 1 plus x) divide by ( minus 1 plus x cubed )
- ( minus one plus x) divide by ( minus one plus x to the power of three)
- (-1+x)/(-1+x3)
- -1+x/-1+x3
- (-1+x)/(-1+x³)
- (-1+x)/(-1+x to the power of 3)
- -1+x/-1+x^3
- (-1+x) divide by (-1+x^3)
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Similar expressions
- (-1+x)/(-1-x^3)
- (-1-x)/(-1+x^3)
- (1+x)/(-1+x^3)
- (-1+x)/(1+x^3)
Limit of the function (-1+x)/(-1+x^3)
The solution
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[src]
/ -1 + x\
lim |-------|
x->1+| 3|
\-1 + x /
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right)$$
Limit((-1 + x)/(-1 + x^3), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x^{3} - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \left(x^{3} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{3 x^{2}}\right)$$
=
$$\lim_{x \to 1^+} \frac{1}{3}$$
=
$$\lim_{x \to 1^+} \frac{1}{3}$$
=
$$\frac{1}{3}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x^{3} - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \left(x^{3} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{3 x^{2}}\right)$$
=
$$\lim_{x \to 1^+} \frac{1}{3}$$
=
$$\lim_{x \to 1^+} \frac{1}{3}$$
=
$$\frac{1}{3}$$
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
One‐sided limits
[src]
/ -1 + x\
lim |-------|
x->1+| 3|
\-1 + x /
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
/ -1 + x\
lim |-------|
x->1-| 3|
\-1 + x /
$$\lim_{x \to 1^-}\left(\frac{x - 1}{x^{3} - 1}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
= 0.333333333333333
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{x - 1}{x^{3} - 1}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 1}{x^{3} - 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{x^{3} - 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{x^{3} - 1}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 1}{x^{3} - 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{x^{3} - 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x^{3} - 1}\right) = 0$$
More at x→-oo
The graph