Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-6-x+2*x^2)/(-2+x)
-
Limit of (sqrt(-1+2*x)-(-2+3*x)^(1/5))/(-1+x)
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Limit of (4+2*n^3)/(5+n^2)
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Limit of (2+4*x)/(-3+2*x)
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Identical expressions
- eight + thirteen *x+ fifty-seven *x^ two / ten
- 8 plus 13 multiply by x plus 57 multiply by x squared divide by 10
- eight plus thirteen multiply by x plus fifty minus seven multiply by x to the power of two divide by ten
- 8+13*x+57*x2/10
- 8+13*x+57*x²/10
- 8+13*x+57*x to the power of 2/10
- 8+13x+57x^2/10
- 8+13x+57x2/10
- 8+13*x+57*x^2 divide by 10
-
Similar expressions
- 8-13*x+57*x^2/10
- 8+13*x-57*x^2/10
Limit of the function 8+13*x+57*x^2/10
The solution
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[src]
/ 2\
| 57*x |
lim |8 + 13*x + -----|
x->oo\ 10 /
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right)$$
Limit(8 + 13*x + (57*x^2)/10, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{57}{10} + \frac{13}{x} + \frac{8}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{57}{10} + \frac{13}{x} + \frac{8}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{8 u^{2} + 13 u + \frac{57}{10}}{u^{2}}\right)$$
=
$$\frac{8 \cdot 0^{2} + 0 \cdot 13 + \frac{57}{10}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{57}{10} + \frac{13}{x} + \frac{8}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{57}{10} + \frac{13}{x} + \frac{8}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{8 u^{2} + 13 u + \frac{57}{10}}{u^{2}}\right)$$
=
$$\frac{8 \cdot 0^{2} + 0 \cdot 13 + \frac{57}{10}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \infty$$
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
$$\lim_{x \to \infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = 8$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \frac{267}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \frac{267}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = 8$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \frac{267}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \frac{267}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{57 x^{2}}{10} + \left(13 x + 8\right)\right) = \infty$$
More at x→-oo
The graph