Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(x)/x+tan(x)
-
Limit of ((1+x)/x)^(1+x)
-
Limit of ((1+x)/(6+x))^(5-5*x)
-
Limit of e^(1/(-1+x))*(-1+x)
- Integral of d{x}:
-
x/(x^2-x)
-
Identical expressions
- x/(x^ two -x)
- x divide by (x squared minus x)
- x divide by (x to the power of two minus x)
- x/(x2-x)
- x/x2-x
- x/(x²-x)
- x/(x to the power of 2-x)
- x/x^2-x
- x divide by (x^2-x)
-
Similar expressions
- x/(x^2+x)
- (2+x)/(x^2-x)
Limit of the function x/(x^2-x)
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^2 - x), 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
$$\lim_{x \to \infty}\left(\frac{1}{x \left(1 - \frac{1}{x}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{1 - u}\right)$$
=
$$\frac{0}{1 - 0} = 0$$
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
$$\lim_{x \to \infty}\left(\frac{1}{x \left(1 - \frac{1}{x}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{1 - u}\right)$$
=
$$\frac{0}{1 - 0} = 0$$
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