![]() Many functions have the virtue of being able to trace their graphs with a pencil without removing the pencil off the paper. Limit does not exist.Definition of Continuity But as we've already seen, just because you can't apply the theorem does not mean that the Limits are not the same here, so this thing does notĮxist, does not exist and so we don't meet thisĬondition right over here, so we can't apply the theorem. Two from the right, it looks like g is approaching zero. When we look at approachingĪpproaching negative two. Well, we first wanna see what is the limit as x approaches two of g of x. Pause this video, we'll first see if this Let's say we wanted toįigure out the limit as x approaches two of f of g of x. 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. So in both of these scenarios, our value of our functionį is approaching zero. Now if we approach two from below, it looks like the value The right, right over here, it looks like the value of our function is approaching two from below. Is approaching zero and then we can go the other way. One way to think about it, when x approaches zero from the left, it looks like g isĪpproaching two from above and so that's going to be the input into f and so if we are nowĪpproaching two from above here as the input into f, it looks like our function For example, in this situation the limit actually does exist. So we do not meet this secondĬondition right over here, so we can't just directlyĬan't apply the theorem does not mean that the limitĭoesn't necessarily exist. ![]() ![]() So when x is equal to two, it does not look like f is continuous. Now let's see the second condition, is f continuous at that limit at two. 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 Well, the first thing to think about is what is the limit as xĪpproaches zero of g of x to see if we meet this first condition. First of all, pause this video and think about whether To figure out the limit as x approaches zero of f of g of x, f of g of x. In this video we'll doĪ few more examples, that get a little bit more involved. Video we used this theorem to evaluate certain types So, while the conditions in the theorem provide a convenient and efficient method for evaluating limits of composite functions in certain cases, they may not always be applicable, and additional analysis or reasoning may be needed to determine the limit accurately. Sal, in the video example, illustrates how he uses his understanding of the behavior of f and g from both the left and right sides of the limit point to deduce the limit of the composite function even though the conditions of the theorem are not met. In such situations, further analysis and reasoning may be required to determine the limit of the composite function. However, in some cases, the conditions of the theorem may not be met, and the theorem cannot be directly applied. If the conditions of the theorem are met, it provides a straightforward way to evaluate the limit of a composite function by first evaluating the limits of the individual component functions and then applying the function's composition. The conditions in the theorem are set to ensure that the theorem can be applied accurately and that the limit of the composite function can be determined correctly. 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. The reason this is confusing because you say 'above' but you come from the opposite direction in the video. Are you referring to from left and right or to literally above the function (meaning the y-values are greater than that point. This whole thing from 'above' and below' is also odd to me. 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. 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.
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