Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of (1+2*n)/(-1+3*n)
-
Limit of (x^3-3^x)/(-3+x)
-
Limit of -x+(-2+x)^4/(3+x)^4
- Graphing y =:
- x/(x-x^2)
- Integral of d{x}:
- x/(x-x^2)
-
Identical expressions
- x/(x-x^ two)
- x divide by (x minus x squared )
- x divide by (x minus x to the power of two)
- x/(x-x2)
- x/x-x2
- x/(x-x²)
- x/(x-x to the power of 2)
- x/x-x^2
- x divide by (x-x^2)
-
Similar expressions
- (2+x^2-3*x)/(x-x^2)
- tan(5*x)/(x-x^2)
- x/(x+x^2)
Limit of the function x/(x-x^2)
The solution
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/ x \
lim |------|
x->oo| 2|
\x - x /
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right)$$
Limit(x/(x - x^2), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x \left(-1 + \frac{1}{x}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x \left(-1 + \frac{1}{x}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{u - 1}\right)$$
=
$$\frac{0}{-1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x \left(-1 + \frac{1}{x}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x \left(-1 + \frac{1}{x}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{u - 1}\right)$$
=
$$\frac{0}{-1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right) = 0$$
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
$$\lim_{x \to \infty}\left(\frac{x}{- x^{2} + x}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{x}{- x^{2} + x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{- x^{2} + x}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{- x^{2} + x}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{- x^{2} + x}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{- x^{2} + x}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{- x^{2} + x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{- x^{2} + x}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{- x^{2} + x}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{- x^{2} + x}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{- x^{2} + x}\right) = 0$$
More at x→-oo
The graph