Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-15+x^2-2*x)/(-15-7*x+2*x^2)
-
Limit of (1/4+1/x)/(4+x)
-
Limit of (-exp(-sin(2*x))+exp(tan(2*x)))/(-1+sin(x))
-
Limit of (e^(4*x)-e^(2*x))/x
- Integral of d{x}:
-
x/(x^3-x)
- Graphing y =:
- x/(x^3-x)
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Identical expressions
- x/(x^ three -x)
- x divide by (x cubed minus x)
- x divide by (x to the power of three minus x)
- x/(x3-x)
- x/x3-x
- x/(x³-x)
- x/(x to the power of 3-x)
- x/x^3-x
- x divide by (x^3-x)
-
Similar expressions
- x/(x^3+x)
Limit of the function x/(x^3-x)
The solution
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/ x \
lim |------|
x->oo| 3 |
\x - x/
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right)$$
Limit(x/(x^3 - x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \cdot \left(1 - \frac{1}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \cdot \left(1 - \frac{1}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{- u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{- 0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \cdot \left(1 - \frac{1}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \cdot \left(1 - \frac{1}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{- u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{- 0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right) = 0$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x}{x^{3} - x}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{x}{x^{3} - x}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{x^{3} - x}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{x^{3} - x}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{x^{3} - x}\right) = \infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{x^{3} - x}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{x^{3} - x}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{x^{3} - x}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{x^{3} - x}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{x^{3} - x}\right) = \infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{x^{3} - x}\right) = 0$$
More at x→-oo
The graph