"Fun" With Limits Early in Calculus

Update: The limit below can easily be derived using L'Hopital's theorem. The purpose of this article is to provide practice in algebra and limit manipulations, limit properties and the definition of the derivative prior to using this theorem.











Important Notes:
(1) The condition n ≠ 0 can be relaxed. At some point, students should be asked to analyze the need for restrictions.
(2) The instructor may well want to avoid giving students the above formula, preferring to have them derive it at least in the positive integer case (see comments below in red under "Developing the Problem").
(3) Note that m and n are not restricted to be positive integers. It is recommended that the instructor begin with this restriction on m and n to allow for an algebraic derivation.
(4) This is not an introductory limit exercise! Please read comments (red, bold) below about starting with concrete numerical values before attempting this generalization.

BACKGROUND/OVERVIEW
This is the time of year when Calculus students quickly move into those wonderful limit problems. Epsilon-delta arguments may not be as popular these days but the mechanics of limits are still the challenge for students. Those who have taught this know that students generally struggle with the algebraic simplifications and procedures. Other than these manipulations students generally feel this topic is easy:

Possible Student Thinking: "You just do some algebra, eliminate the "bad" factor in the denominator and plug in. Easy stuff!"

Naturally, if the assignments contain more theoretical limit problems, they may not feel that way!

On the other hand, the algebra can be a major stumbling block for the more challenging exercises. In this post, I will uncharacteristically deemphasize the theory behind the "cancel and plug in" technique and focus on the algebra at first. Then we will move on to relating the limit to the definition of the derivative and application of some important limit properties, in other words, theory! Of course, if L'Hopital's Theorem were introduced early on (in the chapter on differentiation), that would clearly be the method of choice for students!

DEVELOPING THE PROBLEM (SCAFFOLDING)
If m and n in the limit above are positive integers, students can attempt to factor out "x-a" from the numerator and denominator and substitute x = a into the resulting reduced expression. However, many students struggle with such general factoring formulas (or may not have seen them.) Therefore, synthetic division can be used to generate the other factor. This reviews some nice Algebra 2 but what if m and n are not positive integers? What if they are rational or even irrational? Standard factoring techniques would not apply in general so what to do?

I certainly am not suggesting that the instructor begin with the general problem. In fact, I would 'concretize' the problem using a few special cases:
n=2,m=1
n=3,m=1
n=3,m=2
n=1,m=2
n=2,m=2 (This special case is worthwhile as it reviews basic definitions and limit properties).
n=4,m=5 (requires more sophisticated factoring or synthetic)

Based on these exercises, the instructor may ask students if they can develop a general formula for any positive integer exponents. this is in lieu of giving them the formula at the beginning.

FOLLOW-UP: THE GENERAL CASE
After the definition of the derivative is given, students can attempt the more general version. This is a fairly sophisticated limit manipulation but one worth assigning. I may outline the method in an addendum to this post or in the comments or wait for one of our astute readers to contribute! As a hint, the technique I used is related to the derivation of L'Hopital's Theorem!