Mean Girls Math Problem

You may recall in last week’s newsletter In the acclaimed film Good Will Hunting, I introduced the second math challenge. You might remember the film's assertion that the MIT mathematics department spent two years figuring it out. However, as I demonstrated last week, the problem was relatively simple, and the character portrayed by Matt Damon only managed to reveal 80% of the solution.

This week, we will delve into another film where a math conundrum plays a key role in the storyline. The feature is Mean Girls, where a high school mathematics contest culminates in Cady, portrayed by Lindsay Lohan, confronting the following question:

You can view the specific scene in the YouTube video linked here. The question is raised at the 4:53 mark.

If we were permitted to utilize a spreadsheet, we would clearly observe that as x moves closer to 0 from the negative aspect, f(x) tends to infinity. Conversely, as x approaches 0 from the positive side, f(x) heads towards negative infinity. Hence, with the assistance of computational tools, it's evident that since the function diverges to both negative and positive infinity, it fails to converge to any specific value. Thus, the limit does not exist.

Nevertheless, the characters in the film were not allowed to use any computers. To assist in addressing the query, let’s revisit the concept of L’Hospital’s Rule

This principle essentially states that if the limit of f(x)/g(x) results in a 0/0 form, then lim f(x)/g(x) can be evaluated as lim f\"(x)/g\"(x). Please click on the provided link for additional conditions that I won't enumerate here.

I kindly ask that you don’t reach out to correct me for neglecting to place a circumflex accent over the 'o' in L'Hôpital. I believe both spellings are valid. I’m also uncertain about its pronunciation, so I’m relieved that it’s not a requirement for this newsletter.

Initially, let’s introduce another problem where L'Hôpital's Rule can be applied. This problem is sourced from the Wikipedia page I referenced earlier.

As x approaches 0, both the numerator and the denominator converge to 0. This leaves us at an impasse with the 0/0 indeterminate form. According to L'Hôpital's Rule, we can differentiate both the numerator and denominator, then analyze the limit of that quotient as x nears zero. Let's explore that further.

There we go, L’Hospital to the rescue.

Now, let’s return to analyze the limit from Mean Girls.

We'll employ the product rule for differentiating g(x): Recall that it states (r(x) * s(x))’ = r(x)*s’(x) + r’(x)*s(x).

Unfortunately, applying L'Hôpital's Rule did not resolve the Mean Girls dilemma, leaving us with the unresolved 0/0. Therefore, we can confidently conclude that “there is no limit.” In that scene, Cady was correct, and her team rightfully secured victory in the contest.

I must express my appreciation for Tina Fey, the principal writer of the film, for accurately representing the mathematics! This attention to detail is commendable. As we noted in Good Will Hunting, even major films with substantial budgets can commit mathematical and scientific blunders that truly baffle me.