Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+4*n)/(1+2*n)
-
Limit of (4^n-9^n)/(4^(1+n)+9^(1+n))
-
Limit of (1+1/x)^(-1+x)
-
Limit of 7+x^2-14*x/3
- Graphing y =:
- x-x^2/2
- Factor polynomial:
- x-x^2/2
-
Identical expressions
- x-x^ two / two
- x minus x squared divide by 2
- x minus x to the power of two divide by two
- x-x2/2
- x-x²/2
- x-x to the power of 2/2
- x-x^2 divide by 2
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Similar expressions
- x+x^2/2
- (-x-x^2)/(2+2*x^2+3*x)
Limit of the function x-x^2/2
The solution
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[src]
/ 2\
| x |
lim |x - --|
x->oo\ 2 /
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right)$$
Limit(x - x^2/2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{- \frac{1}{2} + \frac{1}{x}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{- \frac{1}{2} + \frac{1}{x}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u - \frac{1}{2}}{u^{2}}\right)$$
=
$$\frac{-1}{0 \cdot 2} = -\infty$$
The final answer:
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right) = -\infty$$
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{- \frac{1}{2} + \frac{1}{x}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{- \frac{1}{2} + \frac{1}{x}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u - \frac{1}{2}}{u^{2}}\right)$$
=
$$\frac{-1}{0 \cdot 2} = -\infty$$
The final answer:
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right) = -\infty$$
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
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{2} + x\right) = -\infty$$
$$\lim_{x \to 0^-}\left(- \frac{x^{2}}{2} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \frac{x^{2}}{2} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \frac{x^{2}}{2} + x\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \frac{x^{2}}{2} + x\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \frac{x^{2}}{2} + x\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(- \frac{x^{2}}{2} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \frac{x^{2}}{2} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \frac{x^{2}}{2} + x\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \frac{x^{2}}{2} + x\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \frac{x^{2}}{2} + x\right) = -\infty$$
More at x→-oo
The graph