Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of 10+x^2+3*x^3+8*x
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Limit of (-16+2^x)/(-1+5*sqrt(x)*(5-x))
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Limit of (-4+x^2)/(-3+sqrt(1-4*x))
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Limit of ((1+x)/(-2+x))^(3+x)
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Identical expressions
- - two + twelve *x+ twenty-two *x^ two / three
- minus 2 plus 12 multiply by x plus 22 multiply by x squared divide by 3
- minus two plus twelve multiply by x plus twenty minus two multiply by x to the power of two divide by three
- -2+12*x+22*x2/3
- -2+12*x+22*x²/3
- -2+12*x+22*x to the power of 2/3
- -2+12x+22x^2/3
- -2+12x+22x2/3
- -2+12*x+22*x^2 divide by 3
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Similar expressions
- -2+12*x-22*x^2/3
- 2+12*x+22*x^2/3
- -2-12*x+22*x^2/3
Limit of the function -2+12*x+22*x^2/3
The solution
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[src]
/ 2\
| 22*x |
lim |-2 + 12*x + -----|
x->oo\ 3 /
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right)$$
Limit(-2 + 12*x + (22*x^2)/3, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{22}{3} + \frac{12}{x} - \frac{2}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{22}{3} + \frac{12}{x} - \frac{2}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{- 2 u^{2} + 12 u + \frac{22}{3}}{u^{2}}\right)$$
=
$$\frac{- 2 \cdot 0^{2} + 0 \cdot 12 + \frac{22}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{22}{3} + \frac{12}{x} - \frac{2}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{22}{3} + \frac{12}{x} - \frac{2}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{- 2 u^{2} + 12 u + \frac{22}{3}}{u^{2}}\right)$$
=
$$\frac{- 2 \cdot 0^{2} + 0 \cdot 12 + \frac{22}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \infty$$
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
$$\lim_{x \to \infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = -2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = -2$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \frac{52}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \frac{52}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = -2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = -2$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \frac{52}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \frac{52}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{22 x^{2}}{3} + \left(12 x - 2\right)\right) = \infty$$
More at x→-oo
The graph