*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 28^{th} 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 3^{rd} Aaron Samuels asked Cady what day it was, and today is the tenth October 3^{rd} 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.

**Also Read: The record-breaking Nobel Prize winner who pretended he was a gardener**

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

**for every input,**

*f(x)*

*x*So if we consider the function

**x=**3 then

**f(x) =**(sin 3)/3. Sin 3 = 0.05, so

**f(x) =**0.05/3 = 0.17

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

**at which**

*n***will get closer and closer to**

*f(x)***as**

*Lim***gets closer and closer to**

*x***.**

*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:

Which is equal to ∞ for negative numbers approaching 0 and -∞ for positive number approaching 0. [See calculation or click here for a worked video of the proof]

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 3^{rd}_{: }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.

##### **Footnotes**

1. 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.
*******

Matt is sorry that you’re jealous of him, but he can’t help it that’s he’s popular. When he’s not tweeting about buttering muffins on @arcadia_eg0 he can be found writing his PhD at Imperial College.

ChrisThe bottom changes to sin^2 x, and then the limit can be broken into parts… specifically, the second part will be sin x / sin^2 x, for which the limit doesn’t exist as x approaches 0 (think 1 / sin x if you want to make it easier to think about quickly). This seems like something you could do in 13 seconds; although it took me 20 after initially squinting at the screencaps.

MattPost authorThis isn’t mathematically sound:

It was suggested on reddit:

http://www.reddit.com/r/math/comments/2i68mk/would_it_be_possible_to_solve_the_limit_of_the/ckzbkws

And has been floated on other blogs about The Mean Girls limit before

Point 3 here:

http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf450

However I appreciate that one of the points of the article is about if it would be possible to solve this limit in the 13 seconds it takes in the film, and I agree that in a sudden death situation trying something rough and ready to reach an answer that “feels right” is a better approach than plodding through l’hopital’s rule (or, like me, breaking out in a cold sweat at horrified recollections of first year mathematics exams at the mention of the words “Taylor Expansion”)

ChrisYou’re absolutely right, you can’t break down the limit if both individual limits don’t exist. However, I can’t even think of another way to rationalize a 13 second solve.

MattPost authorMe neither: however reddit has educated me that possessing Taylor Expansion Vision when looking at limits is the answer.

phutiGood staff

hof13Forgive me for not having seen the movie (I still haven’t) but I teach math and a student mentioned the movie when I was teaching l’Hospital’s rule and I just got around to watching the limit competition clip. Having taught for 20 plus years, I have another theory about the answer of -1. Assuming these are good students, they would have known the denominator was sin^2 so I will assume they make that change right away. However if they are in a hurry, what they might do is forget the chain rule for both numerator and denominator when using l’Hospitals. That leaves

lim [1/(1-x) – cos x] / sin(2x) , which is still indeterminate.

Another iteration (again forgetting chain) leaves lim [-1/(1-x)^2 + sin x] / cos(2x) = -1

This computation can be done within the time constraints given in the film.

Martin ArgeramiNote that you cannot use L’Hôpital to show that a limit doesn’t exist. The rule is, “if the limit of the quotient of the derivatives exist, then the original limit exists and is equal to it “.

As for solving in a few seconds, it certainly took me less than 13 seconds: for anyone familiar with Taylor expansions, and using that 1-cos^x=sin^2x, the limit is the same as the limit

(-x-x)/x^2=-2/x,

and the limit at zero doesn’t exist. Similarly, the “mistaken” limit with a product instead of a minus would be

-xx/x^2=-1,

even faster than the original version.