How to understand the notion of a limit of a function

["Don't panic!"]

I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize 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.

 

[PeroK]

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!

 

["Don't panic!"]

Looks good to me.

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 $L$?!

All I'd add is that if that limit does not exist at some point x0x_0 then ff is not differentiable at x0x_0. And, such functions do exist!

Good point, I had meant to put that in my description.

 

[PeroK]

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 $L$?!

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 $0$.

 

["Don't panic!"]

Essentially, yes.

Would there be a better way to articulate what I wrote then (or a better way to understand it)?

 

[PeroK]

Would there be a better way to articulate what I wrote then (or a better way to understand it)?

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.

 

["Don't panic!"]

f(x)=xf(x) = x is a simple example of a function that never reaches its limit value at any other point.

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?

 

[PeroK]

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?

I thought you'd already explained that the limit equals the function value precisely when the function is defined and continuous at that point.

 

["Don't panic!"]

I thought you'd already explained that the limit equals the function value precisely when the function is defined and continuous at that point.

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!