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.(adsbygoogle = window.adsbygoogle || []).push({});

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 afixedvalue ##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##convergeto the value ##L## as ##x## approaches the point##x=a##. In other words, thelimitingvalue 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}##convergesto 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.

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

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