Calculus traditionally begins with a look at limits and continuity. In many courses this is the only time students will see discontinuous functions with removable discontinuities and most won’t see jump discontinuities elsewhere. So why do students have to learn about limits so early?

We tell them that they can’t do derivatives without limits. That, of course, is a lie. After a day or so of fussing with calculating derivatives using the limit definition we introduce them to shortcuts and algorithms. In fact, if someone came up to us and asked us out of the blue “what’s the derivative of [insert favorite elementary function here]”, we would not reach for the nearest limit.

We could resort to our favorite standby answer and say “because it’s going to be on the exam”. That doesn’t feel very satisfying though.

For me, limits are all about expectations. They are about predicting the way the world should work if everything is right. As x gets close to a certain value, does f(x) seem to hover at some level or does it fluctuate wildly like sin 1/x as x->0? Does it approach the same value from the left and the right or does there seem to be a jump as there is in f'(x) at 0 for f(x) = |x|? Somehow even |x| seems contrived to some students, but what about f(x)=x^(2/3). Near x=0 what happens to it’s derivative?

Limits let us know we aren’t in Kansas any more.

You certainly need limits if you’re going to prove any interesting theorems – but rather than expecting and entertaining questions like “when are we ever going to need this stuff”, let’s work to get our students to enjoy limits for what they are.

Thoughts?

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This entry was posted on February 18, 2006 at 12:16 am and is filed under Calculus.

February 20, 2006 at 12:14 pm

The thing I like about limits is that it gives you a chance to “feel” calculus. If you jumped into the easy methods of a first derivative, then you really have no idea what’s going on, and it’s very abstract. But limits show you that you really can run across the finish line or be hit with the arrow or whatever those crazy Greeks were thinking at the time.

I think many students tend to forget about the limits, but I think the people who remember have a better chance of being able to use it properly in other parts of their lives.

February 20, 2006 at 3:14 pm

One of the items that was cut from the book (and rightfully so) was Zeno’s Starbuzz Paradox. Where Starbuzz is a large and fast growing chain of coffee houses. The statement is that

You can never arrive at the Starbuzz closest to your home.The proof is, in fact, an early example of a proof by construction. In order to travel to the nearest Starbuzz, you must travel some set distance. This journey takes time. During that time, a new Starbuzz will have been built that was closer to your original location.

April 22, 2006 at 9:55 pm

When do you propose teaching limits? The only other time I can see doing it is between derivates and integrals. You can’t put it off any longer because you need limits to prove the Fundamental Theorem. Nevertheless, I think they are really helpful in explaining the relationship between secant and tangent lines. Since they are such a big part of calculus, they need to go at the beginning.

June 11, 2006 at 3:47 pm

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July 5, 2006 at 4:23 pm

just chanced upon this site. More fun than worrying about limits is playing around with power series, generating functions, formal differentiation etc which give a good payoff for little mental investment. To cope with limits why not adopt the slightly more restrictve definition of differentiability “f(x) diffble at x=a if near x=a its graph lies between two tangent parabolas” (Lipschitz differentiable). Verifying the rules of diffn is easier (you have to do some work, principally on the chain rule for which you need to show that if f and g are bounded near x=a then so is f composed with (x-a)g(x) ). Some functions get excluded (eg x^3/2 at x=0) but we may well not care! The main point is that all those futile questions about limits need not be inflicted on the poor students.

October 9, 2006 at 4:48 am

Derek Hacon’s ideas about Lischitz estimates have been developed rather systematically. The result is a drastic simplification of Calculus. See http://www.mathfoolery.org for details.

January 1, 2007 at 9:29 pm

Limits are important technically (as has been pointed out) and metaphorically, which has also been alluded to, but not explicitly. At issue here is whether space is continuous or discrete, something I don’t think we’ve satisfactorily answered (although I know some think otherwise). I don’t care exactly where we talk about limits, but we should talk about them, if only because it gives us the opportunity to talk about this fundamental question.

October 27, 2008 at 10:33 pm

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April 15, 2009 at 5:36 pm

Hey, cool tips. Perhaps I’ll buy a bottle of beer to the man from that forum who told me to visit your site :)

July 7, 2010 at 8:11 pm

If I had to answer how or why we learn limits, I would say, your best way to understand is to look at a lot of graphs. That is to say, plug functions into the graphing calculator, and ask yourself what the limit is at certain points. First of all, you will understand what a limit is a lot faster than by jumping into methods like reducing ratios, factoring, and applying L’hopital’s rule. Secondly, you will find that you really understand how functions relate to their graphs (a concept many students lack when they go into calculus). You will naturally start to question why we study such tendencies. We tend to draw analogies in order to learn abstract concepts. Before we know it we are saying, “hey this graph looks like something I learned in economics”, or “history”, or “along with that article in the paper”. We ponder why we would want to know a limit in this context. If you can get his far, you figure out why. Its just follow-through at this point.

October 30, 2012 at 2:58 am

Excuse my poor english.

If I had to answer how or why we learn limits, I would say, Limit is a language which represents infinite enlargement of the objects, say the set of real numbers.

May 5, 2013 at 3:47 pm

interesting topic. but as a student I would rather have a tangible word example that starting with limF(x) = x^2+…

to understand and relate limits to real life

September 27, 2015 at 3:02 am

Limit’s are super important to the idea of numerically approximating solutions to differential equations, which is important when applying math to the real world. Without limits, we wouldn’t be able to specify a step size in an algorithm and guarantee a certain degree of accuracy in the solution. Also, if I want to find my speed at any moment, I will calculate the rate of change between two points really close together, knowing (because of limits) that my answer is very close. How close? I can get as close as I need to… because of limits.