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Thread: Limits problem

  1. #1
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    Limits problem

    Hi I want to ask a question: what is the limit of let's say 2/5x. Isn't the limit infinity? That's what I find reasonable, however in a limit calculator it says the limit doesn't exist because it diverges. What does it mean? Thanks.
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    Re: Limits problem

    A limit (if it exists) is a real (finite) number. Therefore no limit is "infinity". We can say that the function f(x)=\tfrac2{5x} grows without bound as x tends to zero from above, and this is what the notational shorthand \lim_{x \to 0^+} \tfrac2{5x}=\infty. Note that "infinity" is not involved at all because real analysis has no "infinity".

    Also note that \lim_{x \to 0} \tfrac2{5x} does not exist because \lim_{x \to 0^+} \tfrac2{5x} \ne \lim_{x \to 0^-} \tfrac2{5x}. The inequality is there for two reasons:
    1. neither limit expression has a finite value at all, so those values cannot be equal;
    2. the left-sided limit expression grows in a negative direction (informally "goes to -\infty", so even in the loosest sense, saying that the limits are "infinity" and "negative infinity" respectively, they are not equal.
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  3. #3
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    Re: Limits problem

    Quote Originally Posted by IloveIl View Post
    Hi I want to ask a question: what is the limit of let's say 2/5x. Isn't the limit infinity? That's what I find reasonable, however in a limit calculator it says the limit doesn't exist because it diverges. What does it mean? Thanks.
    First, learn how to post a readable question. Is it $\frac{2}{5}x\text{ or }\frac{2}{5x}~?$

    Second, the question makes no sense if you do not tell us what x is approaching?
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    Re: Limits problem

    Somehow Archie got it right although I really did forget to include some information. Yes, I asked the question 2/ 5x is approaching where when x is approaching to 0. So if I understand correctly, the limit of 1/x is undefined because the limit diverges when you are approaching from the negative side as opposite to approaching from the positive side. And if I may please explain to me how to write a mathematical expressions here.
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  5. #5
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    Re: Limits problem

    There is no one-sided limit from either side because the function diverges, it grows without bound.

    There is no two-sided because the one-sided limits don't exist (and are therefore not equal). The definition of the two-sided limit is that it is equal to the one-sided limits when both exist and are equal).
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  6. #6
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    Re: Limits problem

    Quote Originally Posted by IloveIl View Post
    Yes, I asked the question 2/ 5x is approaching where when x is approaching to 0.
    Write it with grouping symbols around the denominator, then, such as: 2/(5x).
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