Mister Exam
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How to use it?
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Limit of (19-32*x+9*x^4+16*x^6)/(38-12*x^5+8*x^6+12*x)
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Limit of 7+7*x/3
- Limit of (-1+x^m)/(-1+x)
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Identical expressions
- seven + seven *x/ three
- 7 plus 7 multiply by x divide by 3
- seven plus seven multiply by x divide by three
- 7+7x/3
- 7+7*x divide by 3
-
Similar expressions
- 7-7*x/3
Limit of the function 7+7*x/3
The solution
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/ 7*x\ lim |7 + ---| x->oo\ 3 /
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right)$$
Limit(7 + (7*x)/3, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{7}{3} + \frac{7}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{7}{3} + \frac{7}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{7 u + \frac{7}{3}}{u}\right)$$
=
$$\frac{0 \cdot 7 + \frac{7}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{7}{3} + \frac{7}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{7}{3} + \frac{7}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{7 u + \frac{7}{3}}{u}\right)$$
=
$$\frac{0 \cdot 7 + \frac{7}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right) = \infty$$
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
$$\lim_{x \to \infty}\left(\frac{7 x}{3} + 7\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{7 x}{3} + 7\right) = 7$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{7 x}{3} + 7\right) = 7$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{7 x}{3} + 7\right) = \frac{28}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{7 x}{3} + 7\right) = \frac{28}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{7 x}{3} + 7\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{7 x}{3} + 7\right) = 7$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{7 x}{3} + 7\right) = 7$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{7 x}{3} + 7\right) = \frac{28}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{7 x}{3} + 7\right) = \frac{28}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{7 x}{3} + 7\right) = -\infty$$
More at x→-oo
The graph