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Math Help - Positive or Negative Infinity?

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    Positive or Negative Infinity?

    \lim_{x -> 0} (5x - \frac{4}{x})

    Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?
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    Quote Originally Posted by RogueDemon View Post
    \lim_{x -> 0} (5x - \frac{4}{x}) Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?
    Have a look at this.
    It appears that it depends on the side of 0 we choose.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by RogueDemon View Post
    \lim_{x -> 0} (5x - \frac{4}{x})

    Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?

    The calculator is correct. Take into account that \infty\neq +\infty . Look at the three posibilities: two sided, plus and minus.
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    I don't understand why \infty \neq +\infty.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by NOX Andrew View Post
    I don't understand why \infty \neq +\infty.
    Supposing f defined in a punctured neighbourhood of a:

    \displaystyle\lim_{x \to a}{f(x)}=+\infty \Leftrightarrow  \forall{M>0}\;\exists{ \epsilon >0}:\textrm{\;if\;}0<|x-a|< \epsilon \textrm{\;then\;} f(x)>M

    \displaystyle\lim_{x \to a}{f(x)}=\infty \Leftrightarrow  \forall{M>0}\;\exists{ \epsilon >0}:\textrm{\;if\;}0<|x-a|< \epsilon \textrm{\;then\;}| f(x)|>M

    Our case |f(x)|=|5x-4/x|\to +\infty as x\to 0 (two sided) so, satisfies the second definition.
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    Strictly speaking, that limit does not exist, even as "plus infinity" or "negative infinity"! The two one sided limits are
    \lim_{x\to 0^+} f(x)= -\infty (for example, if x= 0.0001, f(x)= f(0.0001) = 5(.0001)- \frac{4}{.0001} = .0005- 40000= -39999.9995 and
    \lim_{x\to 0^-}f(x)= +\infty (for example, if x= -0.0001, f(x)= f(-0.0001)= 5(-.0001)- \frac{4}{-.0001}[tex]= -.0005+ 40000= 39999.9995.

    I don't understand why -\infty\ne\infty.
    I was sorely tempted to ask if you understood why -5\ne 5!

    Of course, neither " \infty" nor " -\infty" is actually a number. We are really talking about limits of functions or sequences. -\infty and \infty are different because sequence that converge to each, such as 1, 2, 3, 4, ..., n, ..., which has limit \infty, and -1, -2, -3, -4, ..., -n, ..., which has limit -\infty, are clearly getting farther apart so cannot converge to the same thing.

    (From a "topological" viewpoint, adding \infty and -\infty "compactifies" the real number line making it topologically equivalent to a closed interval. Given any topological space, we can always find its "one point compactification", adding a single point, and "open" sets containing that one point, to make it compact- However, it changes the topological properties of the space much more than the "Stone-Chech compactification" above. If we did that with the set of real numbers, we would get a single "point at infinity" but now the it would be topologically equivalent to a circle rather than a line segment.)
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    You seem to have misquoted me. I understand the difference between negative infinity and infinity, just not the difference between positive infinity and infinity.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    Strictly speaking, that limit does not exist, even as "plus infinity" or "negative infinity"!
    When we compactify \mathbb{C} adding only one point to obtain an homeomorphism with S^2 , we denote the added point to \mathbb{C} by \infty .

    When we compactify \mathbb{R} adding only one point to obtain an homeomorphism with S^1, how do you denote the added point?
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