A quick note: I’ll do the “Simple But Evil” series weekly on Sundays — this’ll give us a sense of schedule. If you have topics you want me to write about (or if you want to contribute), send me a note.

L’Hospital’s Rule is a funny topic in the Calculus sequence. For English speaking students, the first hurdle is getting over the fact that it’s not pronounced like “le hospital”. And to be fair, my high school French is probably not enough to correctly pronounce L’Hospital either, but it’s at least in the ballpark.

What makes L’Hospital’s rule funny? By the time we’ve gotten to this in a Calculus II course, enough students have finally gotten it beat into them that $$[\frac{f(x)}{g(x)}{]}^{\prime }\ne \frac{f^{\prime }(x)}{g^{\prime }(x)}$$ and now all of a sudden here’s this new magical rule that allows them to do exactly this! So all that hard work we’ve done as instructors to disavow students of this notion and to get them to accept the quotient rule is ready to get tossed out. Hence, the first thing in simple but evil is to force L’Hospital’s rule questions that will disambiguate the quotient rule and L’Hospital’s rule. Of course, I suggest doing this once we’ve done some of the standard exercises.

Quotital’s Rule

One of at least these four things will happen with this problem
$$\underset{x\to \mathrm{\infty }}{lim}\frac{\frac{1}{x+1}}{e^{-x}}$$

Either

1. Students will rework the function to $\frac{e^x}{x+1}$;
2. Students will have a meltdown reworking the function as per (1) above;
3. Students will be generally confused about how to differentiate the numerator of the “main” fraction;
4. Students will say either that $\frac{\mathrm{\infty }}{\mathrm{\infty }}=1$ or $\frac{0}{0}=1$.

If we see (1), then odds are things are going in a good direction. However, we haven’t been able to see how the work with a quotient rule with L’Hospital’s rule type problem. Hence, we haven’t yet been able to detect if there is a conceptual crack. For this, I suggest this problem or any similar variation:

$$\underset{x\to \mathrm{\infty }}{lim}\frac{\ln (1+\frac{2}{x})}{\sin (\frac{x}{x^2+4})}$$

This problem is a bit more complicated, but students reworking the fraction correctly in the first problem have earned a challenge. All sorts of mistakes are bound to come out — mistakes with differentiating natural log, chain rule with a quotient rule with a trig function, and quotient rule vs L’Hospital’s rule, to name a few.

If we see (2), we’ve got some Algebra / Pre-calc rust that needs to be fixed and L’Hospital’s rule ought to be less of a focus.

If we see (3), then, aha! We’ve found the quotient rule and L’Hospital’s rule crack!

If we see (4), then they haven’t been paying attention. Either that or they don’t want to deal with quotient rule and L’Hospital’s rule and so they’re hoping that they’ll guess their way out of this one.

Exponentials and the Variable of Differentiation

A standard L’Hospital’s rule question is
$$\underset{x\to 0}{lim}\frac{a^x-1}{x},a>0$$

It’s a good question in that we bring back implicit differentiation. Students by now have become over-conditioned on differentiating $e^x$ exclusively that they forget that there exist other exponential functions. But! As in the previous Simple But Evil series, I suggest a bunch of problems grouped together.

1. ${\displaystyle \underset{x\to 0}{lim}{e}^x-1}$ — annoying because it’s not L’Hospital’s rule. You may be surprised at how many students will profess to not knowing how to find this limit all of a sudden. Their brains have been locked into the correlation between limits and derivatives. Break that correlation here.
2. ${\displaystyle \underset{x\to 0}{lim}\frac{e^x-1}{x}}$ — similar to the first problem in the Quotital’s Rule section, but now grouped in with this exercise for a different effect. Most students should get this problem correct.
3. ${\displaystyle \underset{x\to 1}{lim}\frac{x^k-1}{x-1}}$ — another standard problem. Most students should nail this. A bonus exercise is to ask them to do this without using L’Hospital’s rule.
4. Don’t ask ${\displaystyle \underset{x\to 0}{lim}\frac{a^x-1}{x}}$ instead, ask ${\displaystyle \underset{k\to 0}{lim}\frac{x^k-1}{k}}$ — students are over-conditioned on (a) $x$ is a variable, (b) $x$ is the variable of differentiation, and (c) what a polynomial is and what an exponential is as a result of (a) and (b)

The fourth limit will bring to surface many a confusion. So, be patient. Another tactic is to ask the standard question that I say not to ask and then follow it up with my variation. You’ll get similar information about what students know and don’t know, but in increments.

It’s Not A Number!

Students are used to seeing things like $\underset{x\to 0}{lim}$ or $\underset{x\to \mathrm{\infty }}{lim}$. They aren’t used to seeing things like $\underset{x\to a}{lim}$. That’s not to say they’ve never seen a limit where $x$ goes to something other than a finite value or infinity, but more that it’s not often. So, then, what of this?

$$\underset{x\to a}{lim}\frac{x}{\sin (x)}$$

If they’re over-conditioned on the form, then they’ll differentiate via L’Hospital’s rule and get $\frac{1}{\cos (a)}$. If they’re over-conditioned in a different way, then they’ll substitute and obtain $\frac{a}{\sin (a)}$. Neither of these answers are correct. Though, the second is less incorrect. The problem with both of these solutions is that they’re not considering what $a$ is. If $a=0$, then we have our classical limit that we explored in Calculus I. If $a=n\pi$ for integer $n\ne 0$, then we have a different mess on our hands. Piecewise solutions are lost in the shuffle.

Constantly Confusing

Back to constantly confusing, it’s worth tossing in problems like ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{\ln (3)}{\sin (8)}}$. This is an endless source of confusion. There is no $x$ in the function, yet we have to take a limit as $x$ tends towards infinity. It’s a thing that was presumably addressed in Calculus I, but these things need reminding.

It’s Fundamental

The Fundamental Theorem of Calculus is taught in Calculus I and forgotten in Calculus II. Bring it back with integrals that are hopeless!

$$\underset{x\to \mathrm{\infty }}{lim}\frac{{\int }_1^x{e}^{t^2}dt}{x}$$

Indeterminate Subtraction

Classic is this problem $$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n}-\sqrt{n+1}$$ where they trick is to multiply by one via $\sqrt{n}+\sqrt{n+1}$ and the numerator works out nicely and conveniently.

But what of this?
$$\underset{n\to \mathrm{\infty }}{lim}n–\ln (n)$$

or this?
$$\underset{n\to \mathrm{\infty }}{lim}{e}^n–\ln (n)$$

or this?
$$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n}–\ln (n)$$

or this?
$$\underset{n\to \mathrm{\infty }}{lim}\sqrt[3]{n}–\sqrt[3]{n+1}$$

One over one over $x$

Limits involving $x\ln (x)$ I think are fairly well-covered. But some variations on this theme are

1. $\underset{x\to 0^{+}}{lim}{x}^x$
2. $\underset{x\to \mathrm{\infty }}{lim}(1+ax{)}^{\frac{b}{x}}$

These poke at students about the set up of $L=\underset{x\to 0^{+}}{lim}{x}^x$ and the subsequent $\ln (L)$ business — a topic from Calculus I that’s worth some reminding.