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Some basic conceptual doubts about limits

  1. Apr 6, 2014 #1
    (1)If the limit of f(x), as x approaches a point a doesn't exist, and the limit of g(x), as x approaches a point a doesn't exist either, the limit of (f+g)(x) and (fxg)(x), as x approaches a point a can possibly exist?

    (2)
    If the limit of f(x) as x approaches a and the limit of (f+g)(x), as x approaches a both exists, so the limit of g(x), as x approaches a, has to exist?

    Thanks.
     
  2. jcsd
  3. Apr 6, 2014 #2

    jbunniii

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    Try to come up with an example where the bad behavior of ##f## and ##g## is canceled out when you form the sum ##f+g##. Try to come up with another example that does the same for the product ##fg##. Can find an example which works for both ##f+g## and ##fg##?

    ##\lim_{x \rightarrow a} f(x)## and ##\lim_{x \rightarrow a}g(x)## exist, what can you say about ##\lim_{x \rightarrow a} (f+g)(x)##?
     
  4. Apr 6, 2014 #3

    Dick

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    Sure. Can you give examples for (1) and maybe sketch a proof for (2)?
     
  5. Apr 6, 2014 #4
    Thank you very much guys, this returned questions were very clarifying and didatic.
    I imagine for (1) two piecewise functions, f(x):= -1 if x<0; 1 if x≥0, and g(x):=1 if x< 0; -1 if x≥0. As x approaches 0, they are clearly not limited, but (f+g)(x) has a limit, and it is 0 (as seen by the side limits). And for the product (fg)(x) it works the same way, as x aproaches 0 from the left we have a -1, and from the right, -1, even though the functions don't have a limit.
    For question (2), g(x) has to have a limit, because the limit of the sum could be seen as a translation (the limit of the sum is equal to the sum of limits), and if g(x) don't have a limit, but f(x) does, the sum wouldn't have a limit. Is that true? Again, thank you very much.
     
  6. Apr 6, 2014 #5

    Dick

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    Close enuf. g=(f+g)-f. If (f+g) and f have limits then their difference does.
     
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