Mean Girls Math Problem

You may recall in last week’s newsletter I presented the second math problem in the movie Good Will Hunting. You may recall the movie claimed it took the MIT math department two years to solve. As I showed last week, not only was the problem quite easy, but the math genius character played by Matt Damon still only provided 80% of the answer.

This week we will look at another movie that has a math problem as an important plot point. The movie is Mean Girls. In a high school math competition, it all comes down to the character Cady, played by Lindsay Lohan, answering the following question:

You may see the scene in question in the following YouTube clip. The question is posed at the 4:53 point.

If we were allowed to use a spreadsheet, we could easily see that as x approaches 0 from the negative side, f(x) approaches infinity. As x approaches 0 from the positive side, f(x) approaches negative infinity. So, we can see with the aid of a computer that since the function approaches both negative and positive infinity that it doesn’t approach anything in particular. Thus, the limit does not exist.

However, they were not allowed to use computers in the movie. To help with solve the problem, let me remind you of L’Hospital’s Rule

It basically says that if the limit of f(x)/g(x) = 0/0, then lim f(x)/g(x) = lim f'(x)/g'(x). Please see the link above for some other conditions I won’t list.

Please do not write to me and say I forgot to put a circumflexcomb above the o in L’Hospital. I think either spelling is acceptable. I am still unsure how to pronounce it and glad I don’t have to for this newsletter.

Let me first post another problem where the use of L’Hospital’s Rule is appropriate.It is taken from the Wikipedia page I linked to previously.

Both the numerator and denominator approach 0 as the limit of x nears 0. We can’t do anything with 0/0. Per L’Hospital’s Rule we can get the derivative of both the numerator and denominator and take the limit as x approaches zero of that fraction. Let’s do that.

There we go, L’Hospital to the rescue.

Now, let’s get back to the Mean Girls limit.

Let’s use the product rule to differentiate g(x): Recall it says (r(x) * s(x))’ = r(x)*s’(x) + r’(x)*s(x).

So, L’Hospital didn’t help with the Mean Girls problem. Since we still left with the meaningless 0/0, we can safely say that “there is no limit.” In the scene, Cady was right and her team deservedly won the competition.

I would like to give all due credit to Tina Fey who was the main writer of the movie for getting the math correct! This is not something to be taken for granted. As we saw in Good Will Hunting, even the most polished high-budge movie can make mathematical and scientific errors that make the hair on the back of my neck stand up.