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I How to understand the notion of a limit of a function

  1. Apr 7, 2016 #1
    I am trying to explain to someone the formal notion of a limit of a function, however it has made me realise that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any errors/misunderstandings.

    The limit of a function ##f## naturally arises when one wishes to consider the behaviour of a function ##f## around a given point, say ##x=a##, i.e. the values ##f(x)## that it takes near that point. In particular, such a notion is useful if the function itself is undefined at ##x=a##. In such an approach, we study the values that ##f## takes as we allow its input variable ##x## to approach the given point ##x=a##. If the values ##f(x)## get closer and closer to a fixed value ##L## as we allow ##x## to get closer and closer to ##a## regardless of whether we approach ##a## from the left or the right (of ##a##) then we can say that the values of ##f## converge to the value ##L## as ##x## approaches the point##x=a##. In other words, the limiting value of ##f## (i.e. the value that it converges to) is the value ##L## and is written symbolically as $$\lim_{x\rightarrow a}f(x)=L$$ This is the statement that by taking a value of ##x## arbitrarily close (but not equal to) ##a##, we can "force" the value of ##f## arbitrarily close to ##L##. This can be formalised mathematically by saying that if for all ##\epsilon >0## there exists a ##\delta >0## such that, for all ##x## (in the domain of ##f##), ##0<\lvert x-a\rvert <\delta\Rightarrow\lvert f(x)-L\rvert <\epsilon##.

    I then go on to give a particularly important example of why the notion of a limit of a function is useful, namely in defining the derivative of a function at a given point. Indeed, we first consider a function ##f(x)## and its difference quotient ##\frac{f(x+\Delta x)-f(x)}{\Delta x}##. This gives the average rate of change in the value of ##f## with respect to a change in its input ##x## from ##x## to ##x+\Delta x##; equivalently, it defines the slope of the secant line passing through the points ##(x,f(x))## and ##(x+\Delta x,f(x+\Delta x))## of the curve ##y=f(x)##. We then ask what it means to calculate the rate of change at each point along the curve of ##f(x)##. Clearly we can't use the difference quotient at a single point ##x## as its value is undefined (we would have ##\frac{0}{0}##), however we can consider what happens as we allow ##\Delta x## to approach zero (such that the points ##x## and ##x+\Delta x## approach one another). Intuitively, we expect that the secant line (that we introduced earlier) should approach the tangent line to the point ##x##, and as such its slope should approach the value of the slope of the tangent line at that point which we label as ##f'(x)##. Now, if by taking values of ##\Delta x## arbitrarily close to (but not equal) ##0##, we can "force" the value of ##\frac{f(x+\Delta x)-f(x)}{\Delta x}## arbitrarily close to the value ##f'(x)##, independently of whether ##\Delta x## approaches ##0## from the left or right hand side of the point ##x##, then we can say that the value of ##\frac{f(x+\Delta x)-f(x)}{\Delta x}## converges to the value ##f'(x)##, and we can say that its limiting value is given by $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=f'(x)$$ SInce this limiting value (that the value of the difference quotient converges to) is exactly the slope of the tangent line to the curve ##y=f(x)## at the point ##x## and hence describes the rate of change in the value of the function ##f## with respect to a ##x## at the point ##x##, we can therefore define the derivative of the function at a given point as the limiting value of the difference quotient ##\frac{f(x+\Delta x)-f(x)}{\Delta x}##, i.e. $$f'(x)=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
    In taking this approach we have avoided having to resort to the ethereal notion of infinitesimals and undefined quantities.

    Would this be a correct understanding of the situation? I really don't want to convey incorrect information to someone.
     
  2. jcsd
  3. Apr 7, 2016 #2

    PeroK

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    Looks good to me.

    All I'd add is that if that limit does not exist at some point ##x_0## then ##f## is not differentiable at ##x_0##. And, such functions do exist!
     
  4. Apr 7, 2016 #3
    Cheers for taking a look! I'm I correct in saying that the ##L## is the value that the values of ##f## converge to as ##x## approaches a given point ##x=a## and hence the limiting value ##L## of the function is a value that we can make ##f(x)## arbitrarily close to for a suitably arbitrarily close choice of ##x## to ##a##, but in general won't actually be equal to the value of the function at ##x=a##? (Of course, if the function is continuous at a given point then its value at that point is equal to the limiting value of ##f(x)## as ##x## approaches that point).
    Also, is it correct to say that in practice the function, in general, never actually reaches the limiting value ##L## exactly (i.e. is never exactly equal to ##L##), just a value arbitrarily close to ##L##?!

    Good point, I had meant to put that in my description.
     
  5. Apr 7, 2016 #4

    PeroK

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    Essentially, yes. The function need not assume the limit ##L## at any point. Obviously, a constant function has the limit value at every point and a function like:

    ##xsin(1/x)##

    Attains the limit value of ##0## (as ##x \rightarrow 0##) infinitely often in any interval containing ##0##.
     
  6. Apr 7, 2016 #5
    Would there be a better way to articulate what I wrote then (or a better way to understand it)?
     
  7. Apr 7, 2016 #6

    PeroK

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    I would just say simply that a function may or may not attain the limit value. I think examples are better than lots of words.

    ##f(x) = x## is a simple example of a function that never reaches its limit value at any other point. And, with the two above, you have three very different examples of functions that tend to a limit in different ways.
     
  8. Apr 7, 2016 #7
    By this do you mean that ##f(x)=x## never reaches its limit value at any point other than the one that we are approaching in the limit? (i.e. in this case, as ##f(x)=x## is continuous, we have that ##\lim_{x\rightarrow a}f(x)=f(a)##, specifically ##\lim_{x\rightarrow a}x=a##. Of course ##x\neq a## at any point other than ##x=a##, but because the limit is defined then we can make ##f(x)=x## arbitrarily close to ##f(a)=a## by choosing a value of ##x## arbitrarily close to ##a##).

    One last thing (sorry to go on a bit). Just to confirm, the limiting value ##\lim_{x\rightarrow a}f(x)=L## is the value that ##f(x)## is converging to as ##x## gets closer and closer to ##a##, and not necessarily the value that ##f(x)## assumes near to ##a## (or even at ##a##). The notation ##L## labels the limiting value (that ##f(x)## is converging to) and the notation ##\lim_{x\rightarrow a}f(x)## is to explicitly note that this limiting value is the value that the function ##f(x)## is converging to as ##x## converges to ##a##, right?
     
  9. Apr 7, 2016 #8

    PeroK

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    I thought you'd already explained that the limit equals the function value precisely when the function is defined and continuous at that point.
     
  10. Apr 7, 2016 #9
    Sorry, yes I had. I have a bad habit of doubting my understanding and then repeating myself trying to convince myself that I'm wrong! :-\
    I think I just need to be more confident in myself perhaps!
     
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