Why do calculus students learn to think of the derivative as a limit?

I read a similar question on MathOverflow: question link here However, I found that there was too much philosophical discussion going on. Even the chosen best answer didn't really answer the question. So, I thought I'd post this question.

• Limit is what the calculus is all about
– Vim
Jun 24, 2015 at 1:13
• How else would you define it?
– Dan
Jun 24, 2015 at 4:06

Because otherwise they are doing mindless computations without actually knowing what is going on. Think back to your calculus days -- what if your teacher said

"Today we will be discussing derivatives. The derivative of most functions is easy to calculate. For example, $\frac{\mathrm{d}}{\mathrm{d}x}x^2 = 2x$. Yes, it is that easy! What do we do with it? The AP exam overlords tell me you can find the slope of a tangent line, do related rates, and find optimized volumes of boxes!"

And so you go on to do 30 tangent line problems, ace your chapter exam, and score a 4 on the easiest AP exam. But wait! You're a math superstar, so now you take on the challenge of being a math major. A few years later you encounter the legendary first semester real analysis class. Suddenly numbers turn into epsilons and deltas, and any intuition with fast, routine computation becomes as useless as that poem you memorized for English class 9 years ago. You have to understand limits, continuity, derivatives, and integrals now. Meanwhile you get lost in the barrage of logic symbols and inequalities, and proving the non-differentiability of $$f(x) = \begin{cases} x^2 &\mbox{if } x \in \mathbb{Q} \\ 0 & \mbox{if } x \notin \mathbb{Q}. \end{cases}$$ at $x \neq 0$ gives you insomnia. The students who learned (and understood) the limit definition of differentiability will finish the problem with 2 minutes of thinking and 3 minutes of writing.

"Hold up: most calculus students aren't going to be math majors. Why not teach them power rule and chain rule and move on?" Well, that defeats the purpose and novelty of calculus. I remember numerically doing difference quotients in my first calculus course, and the entire class was in awe that dividing by 0.002 in the limit definition didn't explode the answer. My teacher drew $f(x) = x^2$ on the board and about 5 secant lines, each getting closer to the tangent line. From that point on, the class understood the significance of the derivative and the power of the limit process. Instead of doing problems, they were solving them. To divorce the limit definition from the derivative is morally and philosophically wrong -- calculus is the first class students can really ask why things are the way they are.

• Well, heck. i can solve that problem easily. lim x->0 of f(x) = 0^2 = 0. Guess that means I understand limits. ;) Jun 24, 2015 at 1:27
• It takes a bit more rigor than that; either way I used that example for differentiability. EDIT: I changed the example slightly. Jun 24, 2015 at 1:28
• This is really an all-around fantastic answer; I wish I could +1 for each of the many parts I particularly enjoyed. Jun 24, 2015 at 2:37
• This is too good an answer. I like the use of phrase "numbers turning into epsilons and deltas" and the "insomnia" stuff. +1. Luckily I did not have to face all this as I got hold of Hardy's Pure Math during my first interaction with calculus. Jun 24, 2015 at 2:59

First a little philosophy: Contrary to popular belief among laymen, mathematics is not about learning and using a set of rules/operations/tricks.

The primary goal of mathematics is teaching people to reason, to think. Mathematics is never about "you use this formula to solve this problem, don't ask any questions", it is about understanding where does that formula come from and why does it help solve this problem.

And then, if you truly want to understand the derivative, the derivative is a limit....

But more importantly, if you want to understand WHY the derivative solves this problem, then you truly need to understand that the derivative is a limit.

Unfortunately most of the times people use formulas without understanding them, and worse without understanding when they can be used or not. This, or even worse a simple typo in a table or book of formulas to be used in certain situations can easily lead to costly mistakes, all this can be avoided by understanding what you are using.

Now less philosophy: Despite the illusion created in an introductory calculus course, your don't really learn how to derivate many functions. You learn to derivate some simple functions and their combinations, but you only deal with the nicest possible examples. How do you deal with a situation which is just a bit more complex (and many times in physics you run into functions which are even more complicated) like for example:

$$f(x) = \begin{cases} x^2\sin(\frac{1}{x}) &\mbox{if } x \neq 0 \\ 0 & \mbox{if } x=0. \end{cases}$$

What is the derivative at $0$?

Why do calculus students learn to think of the derivative as a limit?

One reasom may be because this limit is defined to be the derivative of the function $f$ at the point $a$:

$\lim_{ \Delta x \rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}$

So it is correct to think so.

Now if you mean why don't they consider the applications of Calculus and hence can't describe what Calculus is used for, then this would be due to focus of the course content and their limited exposure to the applications of Calculus which may require advanced knowledge in a specific filed such as Economics or Electrical Engineering. However, this is in my opinion better than to use the definition in Marriam-Webster's Dictionary.

I think the reason is it kind of helps kill two birds with one stone. While they learn about rates of change, they're also starting to become familiar with limits, because finding a derivative is like finding a value that the output of a function will approach, but may not ever equal. We can define s(x) or m(x) as the slope of a tangent line. That would help even more to transition into talking about limits.

• $f(x) = 3x+2$ always has change in y over change in x of 3. Jun 24, 2015 at 1:07

Because ultimately in the real world, when simulating, you'll have to use some kind of numerical approximation. Most functions out there don't have a closed expression for their derivative. More explicitly, solutions to differential equations can usually only be found using numerical methods which exploit various limit definitions of the derivative. One is also able to quantify the error and rate of convergence.

For what it's worth, the linked MO question was answered primarily by teachers whereas the answers here are mostly by students. So the answers here will probably act more in motivating you but the answers on MO act in the benefit of teaching an entire class on a limited schedule. That's why it probably feels like there's a lot of philosophizing going on. The hard truth is that practically no one really gets calculus the first time around. The limit definition is the canonical definition of a derivative. There are others out there but the limit is surely the most popular one. As many others mentioned here, the limit definition is probably the most versatile: you can essentially figure out if any function out there is differentiable or not. That's a tradeoff.

• "ultimately in the real world, when simulating, you'll have to use some kind of numerical approximation." Will you please post an example or two? Jun 24, 2015 at 1:25
• @moonman239: Pricing options and derivatives in finance, simulating fluid flow, figuring out the lift on an airplane wing, planetary and spacecraft movement, weather simulation, biological processes, pretty much any field where the differential equations are non-linear. Jun 24, 2015 at 2:13
• Whilst your statements are strictly true, they omit the fact that when performing real computations there are often some elements in the system of equations that can be solved exactly and consequently vastly simplify speed and accuracy when solving the full set. Numerical approximation and exact solution go hand in hand. Jun 24, 2015 at 3:50