The Husband and I have an argument that has been ongoing since college. You see, I was telling him about learning how to do limits in my high school calculus class. I developed a technique for understanding what to do when x approaches infinity: Plug in infinity for x, cross off any terms that are added to or subtracted from infinity, and then cancel an things out just like you would for any other number. For example, the limit as x approaches infinity for (2x+1)/x is 2. How do I know? Plug infinity in for x. Drop the 1 and in the numerator, because 1 is nothing compared to infinity. You’re left with 2x/x, or two times infinity divided by infinity. Cancel the infinities, and bam! The answer is 2.
When I told my high school calculus teacher how I was doing these limits, he got all upset and said, “You can’t ‘cancel out’ infinity. Infinity is not a number.”
“But it works every time,” I said. “It’s what you were doing in the examples, you just weren’t saying ‘cancel out the infinities.’”
He cast around for a decent response to this simple fact, and said, “Well, fine, but just don’t say you’re ‘plugging in’ and ‘canceling out’ infinity.” I agreed to not say it aloud, but that is how I do limits and how I will always do limits. Because it works every time! And it makes perfect sense to me.
Enter The Husband. Somehow, limits came up as a topic of our conversation. (This is not so bizarre given that we are both in professions that actually require us to take limits sometimes. I know! People really do it!) And when I told The Husband about my conversation with my calculus teacher, he, too, got all upset.
“You’re teacher was right. You can’t ‘plug in’ infinity,” he said, exasperated. “Infinity is not a number!”
“But it works every single time!” I said. “It’s just semantics, here. That’s what you do too. You cancel out infinity when you have infinity over infinity, you just don’t call it that.”
I tried to demonstrate my point by going through some examples of taking limits, but The Husband just got fed up with me every time I got to infinity over infinity. Eventually, the conversation degraded into name-calling.
To this day, if I want to get The Husband’s dander up, I just mention that I’m planning to “cancel out the infinities."
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3 comments:
Mo -- I had this *exact* same argument with my high school calculus teacher, too. He went on and on for fifteen minutes about how you can't just "cancel" infinities. And I, not wanting to be lectured and figuring there'd be no way for him to tell I'd keep thinking about it the same way, nodded my head like I understood the error of my ways, and then just kept thinking about it the same way.
I had a similar problem in middle school in pre-algebra, balancing equations. The teacher would get mad at me for cancelling a term on one side and subtracting it on the other, instead insisting that I add the negative of the term to both sides. Which does the *same damn thing*, except I have to write more things, never minding the fact that adding the negative to the side where the term orginally started out felt like my announcing to the world that I'm so stupid I need to think up and write out what exactly it is I need to add to something to make it equal zero.
I too, plugged in infinity. Blew the lights out on the entire block, or maybe the neighborhood, or perhaps--the whole city? It doesn't matter much, though, because what's a city compared to infinity. That's right. Nothing!
Still reading and trying to comprehend. --Adam
Let me explain the probelm with canceling.
It does work for stupid people. During the day, we interact with a multitude of stupid people that are not smart enough to grasp what is going on at a detailed level. When you say that things just cancel, there is no stated regard for the operation being performed by the "canceling" term. To make it clearer, let us take the following equation:
4x = y+4
When a "stupid" person believes that they can cancel the 4s, you end up with:
x=y
When you should end up with:
x=(y/4)+1
Smart people know what "canceling" means on a per case basis. Your teacher probably interacted with far more "stupid" people than people like you.
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