Mean Girls is arguably the teen movie of my generation.
I’m not sure if this counts as a valid appraisal of a film’s popular impact, but a site-specific Google search of Buzzfeed alone turns up 7,890 results for posts about the 2004 teen rom-com. If this is compared to various other bits of pop culture, we can see that the 28th highest box office film of 2004 has had a phenomenal cultural impact. It rates just behind the 6-film behemoth that is the Star Wars franchise and well ahead of its rival Clueless. Somehow, in the over-saturated high-school drama genre, Mean Girls made its mark.
If you’ve never seen the film, you’ll be wondering why I’m writing about it today. If you’ve seen the film once or twice, you may also be a little bemused. If, like me, you own multiple DVD copies because you keep one at your parents to have something to watch over Christmas, you’ll know that I’m writing this post today because on October 3rd Aaron Samuels asked Cady what day it was, and today is the tenth October 3rd since the movie’s release.
Also if you’ve read this blog before, you’ll know we like to pick up on mathematics and science in pop culture. In Mean Girls, this isn’t particularly difficult as mathematics is used as the metaphorical weather vane for Cady Heron’s descent-into and subsequent ascent-out-of superficiality.
Cady, played by Lindsay Lohan, moves to the USA and upon entering high school is tempted by the popularity offered by hanging out with the eponymous Mean Girls. 90 minutes of comedy japes involving a botched house party and someone not even going here culminates in the Northshore Mathletes reaching the state final of a math Olympiad, with Cady in sudden death trying to find the limit of this function:
Now, first off I wouldn’t be a proper pedant if I didn’t give my compliments to the small, quiet but gently persistent group of mathematicians online who grumble about the film’s script calling this problem an equation. Things are only equations if they have an equals sign in them. This doesn’t and as such is a function. Keep up the good work, internet mathematics pedants!
The Northshore Mathletes win because Cady’s opponent answers that the limit is negative 1, which is incorrect. Cady then correctly answers that the limit does not exist, winning the prize, getting the guy and spawning a decade’s worth of jokes on tumblr.
But what are limits? When and why do they not exist? Is it plausible that Cady could answer it that quickly whilst having the epiphany that literally underpins the entire plot, her character arc and the moral of the story? Luckily we’re here to help you out.
To paraphrase the Wikipedia article on limits,
A function f gives an output f(x) for every input, x
So if we consider the function
But what if we say x= 0? Then f(x) = (sin 0)/0. Sin 0 = 0, so f(x) = 0/0.
But 0/0 isn’t a number. It doesn’t equal 0, or 1 (or infinity), it just sort of sits there uselessly not telling us anything. Instead we need to calculate the limit of the expression at the point.
A function is said to have a limit Lim, when there is a point n at which f(x) will get closer and closer to Lim as x gets closer and closer to n.
So to calculate the limit to f(x) we can plug in some values for x as it approaches 0. We can see in the table below that as the value we put in for x gets closer and closer to 0, f(x) gets closer and closer to 1.
If we look at this on a graph, we can see that this is also the case.
This answer is defined as the “Limit” of f(x) at the point 0, which in this case is 1. This limit would be written as:
Let’s now return to to the question at hand: the sudden death
equation function in Mean Girls.
If you pause the DVD at the right time, you can see:
There are lots of ways to solve this limit. Like the example above we could just plug in x=±1, x=±0.1, x=±0.01 etc until we were happy we’d calculated the limit. However that is computationally very difficult.
Instead a much easier thing to do is to apply something called L’Hôpital’s rule.1
First, you test the function to see if you can apply L’Hôpital’s rule. You do this by plugging 0 in for x and if you do that you get Lim = 0/0.
This means this limit passes one of the 3 criteria for being allowed to apply L’Hôpital’s Rule.
That means that if the original limit reaches 0/0 then it is the same as the limit of the derivative of the numerator divided by the derivative of the denominator.
If you do this, then the function becomes:
This means that as you approach 0 from opposite sides, the function “diverges”, which means the values of f(x) grow wildly and exponentially further away from one another, until they reach plus or minus infinity. Remember the definition of a limit is that f(x) approaches it. In this example f(x) doesn’t approach anything, meaning that, to quote Cady Heron…
But why did the poor Marymount competitor get negative 1? Obviously it’s very easy as a script writer to put in a wrong answer, she could’ve said anything and be wrong. But there is a very simple mistake which would give you negative 1, ‘discovered’ by readers of Alex Kasman’s blog on the topic of the Mean Girls Limit.
The contributor, Kenneth Wildenhain, points out that when he paused the DVD to take a look at the limit that some distortion made him think the negative sign in the numerator of the limit was a multiplication dot. This completely alters the meaning of the equation, and, with the help of an online limit calculator will resolve down to:
Kenneth goes on to point out that the question was not read aloud and was simply displayed on the projector. Furthermore Caroline Krafft (who seriously needed to pluck her eyebrow, whose outfit looked like it was picked out by a blind Sunday school teacher and who had 99 cent lip-gloss on her snaggle tooth)2 wore glasses and was stood sideways on from the projector screen and could’ve easily misread the negative sign for a dot, giving her the answer negative 1.
The amount of youtube comments you see complaining that the answer *is* negative 1 instead of divergent suggests that misreading the – for a ∙ is not an uncommon problem. In her dedication to accurate mathematics, did the Tina Fey also come up with a plausible error? Whilst I don’t want to enter the weird world of writing math-based Mean Girls fan-fiction, in my mind, this is why Ms Krafft got the incorrect answer.
However this does beg the question as to how in the name of hell Carloine Krafft managed to differentiate both halves of the fraction TWICE and then solve each of them for 0 in 13 seconds? Of course, as L’Hôpital’s rule isn’t necessarily the fastest way to solve limits, she could be really quick at doing Taylor expansions – but I’m very skeptical.
[Update: thanks to u/mmmmmmmike on r/math, I’ve realised I may have demonstrated my very journeyman level of mathematics. Taylor expansions are a much quicker way to go if you know them. Gonna use the “I’m a chemist” defense that I use when confronted with anything harder than calculating the number of moles in a solution.]
Also Cady begins her analysis of the question by stating that the limit diverges, all she has to do is forget Aaron’s beautiful face for a second to remember what was on the board, so by this point she’s already solved the question whilst simultaneously having the epiphany that being a Mean Girl will never make her happy. I’ve never met one person this fast at calculus in my life, let alone two. [But thanks to Reddit, I’ve now met loads.]
But what the hell: these are very minor niggles and this is a film. A film where the jokes are funny, I have mathematically consistent head-canon I can blog about, and Glenn Coco gets 4 candy-grams. So let’s all enjoy the tenth anniversary of October 3rd: watch the film on our many copies of it, double-bill it with Clueless, pretend the direct-to-DVD Mean Girls 2 sequel never happened, bake a cake of rainbows and smiles, eat it and be happy, and remember: the limit does not exist.
Footnotes1. By saying L’Hôpital’s rule is easier, I mean that it’s easier computationally. If you don’t know calculus but DO have a trigonometry calculator left over from GCSE maths it’s definitely easier to just plug in numbers approaching the limit. 2. Really worryingly, I was able to write that out verbatim from memory. This article relied heavily on symbolab.com’s limit calculator because I’m lazy and haven’t got an equation plug-in for wordpress yet, and once again I would really recommend Alex Kasman’s blog of Math in Fiction because it’s a really enjoyable read. All screenshots are copyright of Paramount Pictures.