yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Theorem for limits of composite functions: when conditions aren't met | AP Calculus | Khan Academy


2m read
·Nov 10, 2024

In a previous video, we used this theorem to evaluate certain types of composite functions. In this video, we'll do a few more examples that get a little bit more involved.

So let's say we wanted to figure out the limit as x approaches 0 of f of g of x.

First of all, pause this video and think about whether this theorem even applies. Well, the first thing to think about is what is the limit as x approaches 0 of g of x to see if we meet this first condition.

So if we look at g of x right over here, as x approaches 0 from the left, it looks like g is approaching 2. As x approaches zero from the right, it looks like g is approaching two, and so it looks like this is going to be equal to two. So that's a check.

Now let's see the second condition: is f continuous at that limit at two? So when x is equal to two, it does not look like f is continuous. So we do not meet this second condition right over here.

So we can't just directly apply this theorem, but just because you can't apply the theorem does not mean that the limit doesn't necessarily exist.

For example, in this situation, the limit actually does exist. One way to think about it: when x approaches 0 from the left, it looks like g is approaching 2 from above, and so that's going to be the input into f. And so if we are now approaching 2 from above, here's the input into f; it looks like our function is approaching zero.

Then we can go the other way. If we are approaching 0 from the right, right over here, it looks like the value of our function is approaching 2 from below. Now, if we approach 2 from below, it looks like the value of f is approaching 0.

So in both of these scenarios, our value of our function f is approaching 0. So I wasn't able to use this theorem, but I am able to figure out that this is going to be equal to 0.

Now let me give you another example. Let's say we wanted to figure out the limit as x approaches 2 of f of g of x. Pause this video, and we'll first see if this theorem even applies.

Well, we first want to see what is the limit as x approaches 2 of g of x. When we look at approaching 2 from the left, it looks like g is approaching negative 2. When we approach x equals 2 from the right, it looks like g is approaching 0.

So our right and left-hand limits are not the same here, so this thing does not exist. It does not exist. And so we don't meet this condition right over here, so we can't apply the theorem.

But as we've already seen, just because you can't apply the theorem does not mean that the limit does not exist. But if you like pondering things, I encourage you to see that this limit doesn't exist by doing very similar analysis to the one that I did for our first example.

More Articles

View All
Explained: 5 Fun Physics Phenomena
In my last video, I showed you five fun physics phenomena and asked you how they work. You responded with thousands of comments and some video responses. Well, here are my explanations. Let’s start with the cereal because it seems the simplest, but it tu…
Principles for Success: "The Abyss" | Episode 4
Principles for success: an ultra mini-series adventure in 30 minutes and in eight episodes. Episode four: The Abyss. We progress forward until we encounter setbacks. Whether or not we get out of them and continue forward or spiral downward depends on whe…
How To Get Rich According To Steve Jobs
There are a million ways to make a million dollars, and in this video, we’re looking at one of them. If Steve Jobs were alive today, he would be among the top 10 richest people on the planet. Jobs was known to be a non-conformist, a man focused on buildin…
15 Assets That Are Making People Rich
Assets put money in your pocket; liabilities cost you money. The more assets you have making money for you, the richer you are. This is the fundamental rule of getting rich. But that said, here are 15 assets that are making the rich even richer. Welcome t…
How Politicians Keep Getting So Rich
This is Representative Alan Lowenthal, a Democrat in California. He sits on the House Committee on Transportation and Infrastructure, which on the 6th of March 2020 released this report detailing the preliminary findings from an investigation into the Boe…
Comparing payment methods | Consumer credit | Financial Literacy | Khan Academy
Let’s say that we have decided to buy a television for $499, and we now need to think about how we are going to pay for this $499 television. We know we have many different options, and I’m presenting five of them to you in this video. We could pay with c…