Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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- Limit of (x+2*p)/p^2
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Limit of x^2*(e^(1/x)-e^(1/(1+x)))
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Limit of (-9+x^2)/(1+3*x)
- Factor polynomial:
- x+x^2/2
-
Identical expressions
- x+x^ two / two
- x plus x squared divide by 2
- x plus 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
-
Similar expressions
- (-2+x+x^2)/(2+x^2+3*x)
- x-x^2/2
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
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{2} + x\right) = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{2} + x\right) = \frac{3}{2}$$
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
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{2} + x\right) = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{2} + x\right) = \frac{3}{2}$$
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