Frequent calculus error: replacing interior part of an expression with its limit

Question

For example

$$\lim\limits_{n\to\infty}\left(1+\frac{1}{2n+1}\right)^{n} =\lim\limits_{n\to\infty}{1}^{n}=1\,.$$
Here the student has replaced the sub-part $\frac{1}{2n+1}$ with its limit $0$, but he left the other $n$ at the exponent. I always tell them this is wrong, and give examples, but some of them keep making this errors again and again. So I'm interested in other ways to make them avoid this error.

Of course, there are cases when this doesn't lead to errors, if the sub-part which is replaced with its limit can be separated from the rest of the expression, such as in $$\lim\limits_{x\to 1}\frac{\sin(x^2-1)}{x^2+x-2}=\lim\limits_{x\to 1}\frac{\sin(x^2-1)}{x^2-1}\frac{x^2-1}{x^2+x-2}=\lim\limits_{x\to 1}\frac{x^2-1}{x^2 +x-2}\,.$$

I tell them, "here we applied the theorem that says: limit of product $=$ product of limits," but in the above example there is no theorem that can justify it.

Answer 1

Perhaps this is not a good answer per-se, insofar as it yet-once-again questions (to a certain degree) the premises of the question...

But, indeed, in many propitious situations, everything in sight is continuous, and it is justifiable to replace sub-expressions with their limits. I might claim that the practical success of calculus substantially depends exactly on the fact that "doing the obvious thing" very often succeeds in practice. Indeed!

In that context, it is mildly perverse (I do claim) to make beginners worry about bad possibilities that (with luck) do not occur at an entry level. Further, I would claim that the physical intuition that students have (naive as it is) is fairly correct about good limits.

I tell my first-year calculus students that if plugging in the limit value produces a sensible value, it is very likely that it is correct. And I also tell them that if their or other peoples' lives depended on it, they'd want more assurance than that.

So, seriously, what bad limits will people have reasonable motivation to worry about? I think the difficulty many beginners have in taking "the rules" seriously is that all they see are contrived examples quite-obviously designed to "punish" the non-conformist.

That is, the success of calculus and such is that mostly it does work as one might imagine, and even better. It was (as I've ranted many places) not a big hit because of invitations to fretting about rigor, but because it decisively answered questions about real things (whether in the external physical world or in the more-internal mathematical world).

And, for example, I cannot bring myself to advocate "conformance to rules", but I can easily advocate "viewpoint that explains things". Yes, it is not easy to find subtle mathematical issues that have much sense or moment for scientifically naive kids. I conclude that subtle analytic issues should be delayed until they might genuinely matter.

Answer 2

Very good question. I find it helpful to point out common errors such as this. If you make a point to show them the common mistake, they are less likely to continue that mistake. It seems by your post that you've talked to them a little about this, but it may be helpful to go over several common errors that you see in calculus at the beginning of class. For example, many students make the mistake of dropping the limit notation when simplifying. They may understand what they are doing, but this is NOT allowable. I think it is important to talk about mistakes such as these. As for other ways to make them avoid making this error, I'm not sure.

Answer 3

Sometimes a gauntlet of true/false questions on specific (simple) manipulations like this gets some traction. Anytime I see a student mistake like this in class I try to turn it into a true/false question on the board so we can check in with the whole class on their understanding. The true/falsity emphasizes that this is not just some notational nicety.

Answer 4

I like to talk about a "tug of war" or "competing tendencies".

For example, with $\lim_{x \to \infty} (1+\frac{1}{x})^x$, the base of the exponential is going to 1, while the power is tending to infinity. One tendency is driving the whole to 1, and the other to infinity (since the base is always slightly bigger than 1). These two competing tendencies must be resolved in some way before we just "substitute" part of the expression.

On the other hand for an expression like $\lim_{x \to 2} \frac{x+1}{x+2}$, there is no issue. There is no competition here. You can just go ahead and substitute.

Something like $\lim_{x \to \infty} \frac{x+2}{2x+3}$, on the other hand, does have a tug of war. The numerator is "pulling the fraction to infinity" and the denominator is "pulling it to 0". So it would not be fair just to substitute into one part of the expression: that would be declaring a winner in this contest without investigating who is really stronger. So you need to do some more algebra to manipulate this into a form where you can "clearly" see the winner. Rewriting it as $\lim_{x \to \infty} \frac{1+2/x}{2+3/x}$ erases the tug of war. Now you can plainly see the answer is $\frac{1}{2}$

Answer 5

Although they surely can't put it in those terms, this sort of manipulation can be described in terms of multivariable limits; in many situations you can do what they are doing. For example,

$$ \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} = \lim_{(x,y) \to (\infty, \infty)} \frac{1 + \frac{1}{x}}{1 - \frac{1}{y}} = \lim_{x \to \infty} \lim_{y \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{y}} = \lim_{x \to \infty} 1 + \frac{1}{x} $$

I won't try and state the conditions under which this sort of thing is valid since I'll surely get the fine detail wrong; my point is just to show that it's a reasonable and even rigorously justifiable manipulation in sufficiently well-behaved cases.

Because of this, I believe that simply telling them "that's wrong" is not particularly effective.

I think the right approach is two-fold:

• Demonstrate signs that should trigger red flags in their mind; e.g. in the example problem, having a $1^\infty$ form should immediately trigger skepticism about previous steps, and compel them to use more detail, precision, and rigor when they would otherwise not
• Emphasize alternative ways of doing the sort of thing they're trying to do.

For example, the usual trick for rigorously implementing the sort of idea I use above is

$$ \lim_{x \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} = \lim_{x \to \infty} \frac{1}{1-\frac{1}{x}} \cdot \left( 1 + \frac{1}{x} \right) $$

with the key idea being using factoring to pull out the part of the limit I wanted to simplify, and then use limit-of-products to justify it. Maybe a less trivial example is how to replace $\sin x$ with $x$:

$$ \lim_{x \to 0} \frac{\sin x}{\sqrt{1 + x^2} - 1} = \lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{x}{\sqrt{1 + x^2} - 1} $$

While I'm sure I saw these sorts of manipulations a lot in class, it was always "now I do this magical trick that is useful in this case" — I had to develop the underlying goal-oriented strategy myself.

I imagine it would help to explicitly demonstrate it in that fashion. Have an entire lesson or so dedicated to recognizing parts of a limit that one might want to simplify and the algebraic tactics that allow you to enact that simplification.