Day 15: L'Hôpital's Rule

When your students don't know many derivatives, there is only so much you can do with L'Hôpital. I get that, but I'm stubborn. To me, it is incredibly useful for student to know L'Hôpital as soon as possible, so I always taught it in Unit 1 (and revisited it second semester). How was it useful? MANY REASONS!

1.) Explaining limits like x-->0 of sinx/x and x-->0 of (cos(x) - 1)/x is a breeze with L'Hôpital and a pain with Squeeze Thm

2.) Students have a backup method for polynomials encase they mess up factoring or synthetic division.

3.) They have a back up method for limits that approach infinity encase they don't understand which functions increase faster than others

4.) When we're in the thick of derivatives they don't need that one more thing to add to their stress, and the limits unit is easier than the derivatives unit for most students.

5.) Most of the limits they see on the AP exam are l'Hôpital or speed problems and so focusing on that fact early makes sense to me.

6.) I just like having all the limits problems together in one unit.

If I haven't convinced you, that's okay. This blog is really just me telling the reader what me, myself, and I do to teach this class. You do you.

Anyway, so my students only know the derivatives of sin(x), cos(x), e^x, and how to take the derivative of a polynomial at the time of the lesson, so the L'Hôpital questions that they actually work on can only involve those functions. I can still ask them questions about other functions: like does the lim as x-->1 of the function ln(x)/(x - 1) give you an indeterminate form where using L'Hôpital is applicable? I just can't ask them to work those out. Which reminds me, make sure they know that L'Hôpital is for fractions only 0/0 and ∞/∞ (could be plus or minus infinity), some kind of infinity over some kind of infinity. Also make sure their notation is good throughout their work (they like to forget write lim each time). Also make sure that they understand once L'Hôpital gives you an answer, THAT IS YOUR ANSWER. You can't just keep going with taking derivatives of the top and bottom as many times as you want.

My students love this lesson. I've been told that I should have started Unit 1 with it, lol.