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Theorem for limits of composite functions: when conditions aren't met

Suppose we are looking for the limit of the composite function f(g(x)) at x=a. This limit would be equal to the value of f(L), where L is the limit of g(x) at x=a, under two conditions. First, that the limit of g(x) at x=a exists (and if so, let's say it equals L). Second, that f is continuous at x=L. If one of these conditions isn't met, we can't assume the limit is f(L). Created by Sal Khan.

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  • mr pink red style avatar for user telma.toral1
    Can someone please help verify my work? I want to show that f(g(x)) as x->2 does not exist.

    *If...*
    lim(g(x)) as x->2+ = 0.
    from the graph of g, the values of g(x) are approaching 0 from "above" (This will be the input into f) see video @
    so if we approach 0 from "above" in f(x) we can see that the function f(x) is approaching 3 from the right. This is another way of saying lim f(x) as x->0+ = 3

    *And if...*
    lim(g(x)) as x->2- is -2.
    from the graph of g, the values of g(x) are approaching -2 from "above" (This will be the input into f) So we need to do lim f(x) as x->-2+ = 2

    *This means that ...*
    because the two limits are not the same (2 and 3) f(g(x))'s limit does not exist.

    Please let me know if I'm correct/where I messed up!
    (84 votes)
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  • blobby green style avatar for user Adrian Johnson
    why does he say "approaching 2 from above" when he is approaching from below on the f(x) graph, and vice versa a few seconds later?
    (34 votes)
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    • blobby green style avatar for user voxuanquynh2904
      I suppose g(x) to be another variable, for example g(x)=X. So lim f(g(x)) become lim f(X). As the x in g(x) graph approaches 0 from the left, g(x) approaches 2 from above, which means X in f(X) graph approaches 2 from the bigger side (the right side). When X approaches 2 from the right, its graph f(X) (or f(g(x)) as I've supposed before) approaches 0 from below
      Hope this help!
      (8 votes)
  • blobby blue style avatar for user Zoe Tang
    I'm very confused by the whole "from above" and "from below" technique. If by going "from above" Sal means that he's going from a higher point to a lower point on the y-axis when he is finding the limit when x approaches zero for g(x), he doesn't seem to be doing the same thing when he's finding f(2). When he was finding the limit of f(x) when x approaches 2, he was moving from below on the y-axis even though he said was going "from above."
    (31 votes)
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  • blobby blue style avatar for user Isabella Mathews
    So basically what I understood from the video is, if the first condition is true (i.e., limit of g(x) as x-->a exists = L) but the second (f(x) being continuous at L) isn't, then I gotta find the limit of f(x) as x-->L. And if the first condition is not true, then the limit wouldn't exist at all, because there's no L to begin with.

    Have I gotten it right? Any help would be appreciated.
    (10 votes)
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  • blobby green style avatar for user michael.babcock0610
    the constantly changing terminology does not help with the explanation. Switching from to the right of, or to the left of; to above or below is very confusing. Anyone have a better explanation?
    (18 votes)
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  • blobby green style avatar for user taherhaider90
    Math gets nuts beyond this point
    (17 votes)
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  • blobby green style avatar for user Rob Crane
    With all due respect, this video needs to be reworked. I've rewatched portions of it many times. The transition from left to right (even though that's a lower to higher value, does not easily translate to above and below. Yes, I finally get it, but really, this is a much more difficult concept to apply than the 40 seconds allotted to it. You are in effect rotating the graph. Perhaps visualizing that might help. And, while I'm at it, I think you should start this series of lectures on limits with a a discussion of WHY limits are critical to calculus - what function they serve. I'm 74, and have been a teacher most of my adult life (albeit medicine rather than math - which I last took 56 years ago) but I do recognized that "why" is a powerful motivator to understanding. Rob Crane, MD
    (14 votes)
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  • starky tree style avatar for user hmc
    I have watched this video multiple times and I am really unclear on the process of what to do if the conditions are not met.

    So, in this example where the second condition isn't met, the limit of f(g(x)) as x->0 equals 0, right? Is this because g(0) = 2, therefore f(2) = 0, therefore the limit of f(g(x)) as x->0 equals 0? This is what the answer ends up being if you do it that way, but I'm not sure if that's a valid technique or just a coincidence that it lands on 0. This whole thing from 'above' and below' is also odd to me. Are you referring to from left and right or to literally above the function (meaning the y-values are greater than that point... not the x-values). The reason this is confusing because you say 'above' but you come from the opposite direction in the video.

    I understand the theorem, but if I'm understanding this video correctly, is the whole point here that you can just still solve the limit with other methods? And if so, can somebody please clearly explain what the steps are that Sal is doing to solve it in this case? This doesn't seem like an actual process, more just like he is kind of deducing the answer using clues and not solving the problem using a method.
    (11 votes)
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  • blobby green style avatar for user Jillian Zhu
    What's the point of the "theorem" if you can bypass it? Also, what's the theorem he's talking about called?
    (7 votes)
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    • leaf green style avatar for user kubleeka
      A theorem is just a proven statement that says "if these conditions are met, then this result follows." Sal is demonstrating the converse: if these conditions aren't met, then this result doesn't follow. That doesn't invalidate the theorem.

      Also, the theorem he's talking about doesn't have a common name like "Rolle's theorem" or "the Pythagorean theorem" do.
      (5 votes)
  • starky sapling style avatar for user Kaison Toro
    does anyone know where i can find a proof of this theorem?
    (8 votes)
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Video transcript

- [Tutor] 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 zero of f of g of x, 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 zero 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 zero from the left, it looks like g is approaching two, 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 zero from the left, it looks like g is approaching two from above and so that's going to be the input into f and so if we are now approaching two from above here as the input into f, it looks like our function is approaching zero and then we can go the other way. If we are approaching zero from the right, right over here, it looks like the value of our function is approaching two from below. Now if we approach two from below, it looks like the value of f is approaching zero. So in both of these scenarios, our value of our function f is approaching zero. So I wasn't able to use this theorem, but I am able to figure out that this is going to be equal to zero. Now let me give you another example. Let's say we wanted to figure out the limit as x approaches two of f of g of x. Pause this video, we'll first see if this theorem even applies. Well, we first wanna see what is the limit as x approaches two of g of x. When we look at approaching two from the left, it looks like g is approaching negative two. When we approach x equals two from the right, it looks like g is approaching zero. So our right and left hand limits are not the same here, so this thing does not exist, 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.