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### Course: AP®︎/College Calculus AB>Unit 1

Lesson 5: Determining limits using algebraic properties of limits: limit properties

# 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.

## Want to join the conversation?

• 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!
• I got the same exact thing I think you're absolutely correct =)
• 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?
• 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!
• 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."
• when watching the video it looked liked it was the opposite. When Sal said above he came from a higher x value. I to am confused on why he did that
• 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.
• 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?
• Math gets nuts beyond this point
• 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
• Please see the next video to understand what is really happening.
Do not panic quickly when you don’t understand something.
• 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.
• Hi, I think that basically what he's doing is calculating the limit of f(L). So, maybe, we can say lim[f(g(x))]=M for x-> a is the same as lim[g(x)]=L for x-> a and then lim[f(x)]=M for x->L
(1 vote)
• What's the point of the "theorem" if you can bypass it? Also, what's the theorem he's talking about called?