Mister Exam
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How to use it?
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Limit of ((3+5*x)/(-2+4*x))^(1+3*x)
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Identical expressions
- x- two *y
- x minus 2 multiply by y
- x minus two multiply by y
- x-2y
-
Similar expressions
- x+2*y
- y/(3+x)-2*y^(2*x)*log(y)
Limit of the function x-2*y
The solution
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lim (x - 2*y) x->oo
$$\lim_{x \to \infty}\left(x - 2 y\right)$$
Limit(x - 2*y, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x - 2 y\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(x - 2 y\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{2 y}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{2 y}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{- 2 u y + 1}{u}\right)$$
=
$$\frac{- 0 y + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x - 2 y\right) = \infty$$
$$\lim_{x \to \infty}\left(x - 2 y\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(x - 2 y\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{2 y}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{2 y}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{- 2 u y + 1}{u}\right)$$
=
$$\frac{- 0 y + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x - 2 y\right) = \infty$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x - 2 y\right) = \infty$$
$$\lim_{x \to 0^-}\left(x - 2 y\right) = - 2 y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x - 2 y\right) = - 2 y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x - 2 y\right) = 1 - 2 y$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x - 2 y\right) = 1 - 2 y$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x - 2 y\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x - 2 y\right) = - 2 y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x - 2 y\right) = - 2 y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x - 2 y\right) = 1 - 2 y$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x - 2 y\right) = 1 - 2 y$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x - 2 y\right) = -\infty$$
More at x→-oo