# The Physics Philes, lesson 25: Taking Math to New Heights

*In which I do less than promised, a pinky swear is offered, and the universe does not explode.*

Ermahgerd, you guys. This week was rough. It took me a while to understand some things so I didn’t get as much new info crammed into my brain as I wanted to. Which means that I’ll have to drag this whole limit topic out a week longer than I had anticipated. But hey, what can you do? Sometimes it just takes longer to grasp some concepts than others. Anyhoo…let’s get started!

This week I’ll start on the relationship between limits and infinity. We already know that where there are asymptotes, there is no limit. We know that asymptotes just increase or decrease infinitely. So how can we tell if the function we’re dealing with is an asymptote? Don’t worry. There is a method.

As I’m writing this post, the method seems pretty easy. So easy, in fact, that I don’t know why I had such a hard time grasping it in the first place. But it did, which means that this week I’m only going to go over how to evaluate vertical asymptotes.

Any nonzero number divided by zero indicates that the function is increasing and decreasing without bound, aka is a vertical asymptote. (0/0 typically means a hole exists in the graph of that function.) I know, I thought the universe exploded if you divided by zero too. But we’re still here! Let’s do a quick example to show how this works.

At what value of x does no limit exist for:

First, factor. (Seriously! Factoring is everywhere!) We need to find out what x-value will cause a zero in the denominator. This is very important.

Now our equation looks like this:

We need to figure out what value will give us a zero in the denominator, but a nonzero number in the numerator. Let’s check out x = 5.

Ta da! It seems pretty simple, right? All you have to do is figure out how to get a zero in the denominator. But I’m still confused by one thing. How can we tell if the function is going infinitely up or infinitely down?

According to my book, all I have to do is plug in an x-value slightly to the right or slightly to the left. It says that if I plugged in x = 5.00001 that I should come up with f(5.0001) ≈ 700,000. But I don’t get that at all. Not even close. I guess I’m doing something very wrong, but I can’t figure out what. I’ll continue working on it this week, but if you have any insight, please leave it in the comments.

I’m sorry I can’t finish up limits today. I really wanted to (really). Next week, though. Next week I’ll dive right into limits and horizontal asymptotes and I’ll take a look at some special limit theorems. Pinky swear.

*Featured image credit: Shockerz bugs*

Easier way to examine it would be:

(x+5)/(x+5)=1

So all that is left is:

(x+2)/(x-5)

And so:

(5.00001+2)/(5.00001-5)=7.00001/0.00001=700001

I plugged it in directly too and it worked. So I don’t know what you did wrong. Can you show me the step by step arithmetic you did?

BTW, you can also use logic to know how the graph is shaped too. First of all, as x goes increasingly large, the number part becomes irrelevant, so:

(x+2)/(x-5)->x/x

Now, if it goes to negative infinity, then:

-x/-x=1

Also, at x=0, f(0)=-2/5, which means that from x=negative infinity, where f(x) approaches 1, it goes downwards, and since the asymptote is at x=5, f(5) approaches negative infinity as you apporach from the left.

At positive infinity, x/x=1. And when it approaches x=5, it has to be positive because any number above 5 will be:

(above 5+2)/(above 5-5)=positive number/positive number

And so from the right, the curve goes up and f(5) approaches positive infinity.

And so the graph will look like two separate U curves curves, one from the right at x=1 and slowly going up until it gets close to x=5, which then quickly goes up. The other is from the left at x=1, slowly going downwards, intersecting x=-2 and y=-2/5, and then when it gets close to x=5, quickly going downward.