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Behaviour of limits and their effect on equations

  1. Jun 24, 2013 #1
    Hi,

    I was wondering if some one could check my understanding of limits please.

    If a limit is presetned as say x << y or x >> y am I right in thinking that x or y can be ignored as they are small enough to be insignificant? So, for example, if I had equation which was

    sin(x/y)

    in the situation where x << y only values of y are significant and where x >> y then only values of x are significant.

    Or do I have this completely wrong??

    Cheers for any advice
     
  2. jcsd
  3. Jun 24, 2013 #2

    Office_Shredder

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    Staff Emeritus
    Science Advisor
    Gold Member

    You have it wrong.

    If you have an addition, for example 2x+3y and you are told x<<y then the x is so small that it is insignificant and you can approximate this by 3y and be OK.

    But if you have a ratio like x/y, then being told x<<y doesn't let you make any simplification. If you try throwing away the x and writing it as 1/y then you're probably off by orders of magnitude which isn't a very good thing.

    Sample calculations: If x<<y approximate the following:
    1) [tex] \frac{2x + 3y}{4x - y} [/tex]
    In this case the numerator is approximately 3y and the denominator is approximately -y, so this is going to be approximately -3.

    2)
    [tex] \frac{ 2xy}{y} [/tex]
    The answer is obviously 2x. If you throw away the x because it's a lot smaller than y, you'll end up getting 2, which is not close to 2x unless x happens to be close to 1.
     
  4. Jun 24, 2013 #3

    Mark44

    Staff: Mentor

    From what you've written, I don't think you're asking about limits, but instead are asking about how to approximate expressions in two variables, and it's given that x << y or y << x. Note that sin(x/y) is NOT an equation. An equation has an = symbol in it.

    The following is an example of a limit:
    $$\lim_{x \to 0}\frac{\sin (x)}{x}$$

    It's possible to have limits in which a point (x, y) is approaching some fixed point (x0, y0), but it's very unusual for it to be given that x << y or the other way around.
     
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