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# Limits of composite functions

AP Calc: LIM‑1 (EU), LIM‑1.D (LO), LIM‑1.D.1 (EK), LIM‑1.D.2 (EK)

## Video transcript

- [Voiceover] Let's now take some limits involving composite functions. So over here we have the limit of g of h of x as x approaches three, and like always, I encourage you to pause the video and see if you can figure this out on your own. Well, we can leverage our limit properties, we know that this is going to be the same thing as the limit, actually let me write it this way, this is going to be the same thing as g of the limit as x approaches three of h of x, or I could say the limit of h of x as x approaches three. And so we just need to figure out what the limit of h of x is as x approaches three. So, let's look at h of x right over here and as x approaches three, so we see that h of three is undefined but we can think about what the limit of h of x is as x approaches three. As x approaches three from the left, as x approaches three from the left, we see that the function just stays at a constant two, so at h of 2.5 is two, h of 2.9 is two, h of 2.999999999 is two. So it looks like when we approach from the left, the limit is two and we approach from the right, we get the same thing, h of 3.01 is two, h of 3.001 is two, h of 3.0000001 is two. So this limit right over here is two. So this is all simplified to g of two. Now what is g of two? Well, let's see, this function here, when x is two, g of two is zero, so this right over there is going to be zero, and we're done, let's do a few more of these. Alright, so we wanna find the limit as x approaches negative one of h of g of x. Well, just like we just did, this is going to be the same thing, this is equal to h of the limit as x approaches negative one of g of x, so let's try to figure out the limit of g of x as x approaches negative one. So, this is the graph y equals g of x and we see at negative one right over here, we have this discontinuity and as we approach, as we approach x equals negative one from the left, it looks like we go unbounded in the negative direction, so you could say we're approaching negative infinity, and as we go from the right, as we go from the right, looks like we are, as we get closer to x equals negative one on the right-hand side, looks like we're approaching infinity, so even if they were both approaching the same direction of infinity, we would say that the limit's not defined or at least that's the technical idea here. But this is going, one's going towards positive infinity, and the other is going to negative infinity so this limit right here is undefined. So, it doesn't exist, or I should say, does not exist. So, if the limit as x approaches negative one of g of x does not exist, well, there's no way we that we can evaluate this expression. We can't find h of does not exist, so this entire limit does not, this entire limit does not exist. Let's do one more of these. Alright, so we have once again limit of h of f of x as x approaches negative three, this is the same thing, this is equal to h of the limit as x approaches negative three of f of x. So, let's look at f of x, this is the graph y equals f of x, and the limit as x approaches negative three, well, as we approach negative three from the left-hand side as we get closer and closer to negative three, it looks like we are approaching the value of one, and as we approach from the right-hand side, it looks like we're approaching the value of one. If I were to take from the left-hand side, if I were to take negative 3.1, negative 3.01, negative 3.001, I'm gonna get closer and closer to, if I evaluate the function there, so I should take f of negative 3.1, f of negative 3.01, f of negative 3.0001, we're getting closer and closer to one, and same thing on the right-hand side, so this thing looks like it's one. So, now we just have to evaluate, and I'll rewrite it, so this is the same thing as h of one, h of one, so you just have to evaluate this. Then when we look at this graph here at one, this function does not look defined, so h of one is actually undefined, undefined right here, so also in this case, this limit would not exist. Once again, the limit part was actually at least the limit of f of x was fairly straight forward, but then when we tried to take that output and put it as the input into h of x, well, h of x, h wasn't defined there.
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