![]() Given this restriction, it is clear that all the resulting values of f g (x) are positive. The range of this composite function consists of all values obtained when making the argument of the logarithm positive. So this met the conditionsįor this theorem and we were able to use the theorem to actually solve this limit.Let $A=B=C=\mathbb$ and solve it for $a$:įINALLY, observe that you got the same thing both ways. To find the range of f g (x), we have to identify the function f g (x) first. It looks like at that point, our function is definitely continuous and so we could say that this limit is going to be the same thing as this equals f of the limit as x approaches negative three of g of x, close the parentheses and we know that this is equal to three and we know that f of three is going to be equal to negative one. Is is our function f continuous at this limit,Ĭontinuous at three? So when x equals three, yeah, That first condition and then the second question So this thing is going to be three, so it exists, so we meet As we approach it from either side, the value of the function is at three. Negative three is negative two, but it's a point discontinuity. Suppose (unrealistically) that it stays spherical as it melts at a constant rate of. But let’s start by looking at a simple example of function composition. So it looks like this limit is three, even though the value g of When the functions are linear transformations from linear algebra, function composition can be computed via matrix multiplication. Is actually at three and it looks like when we'reĪpproaching negative three from the left, it looks like Negative three from the right, it looks like our function ![]() Three of g of x, what is that? Well, when we're approaching Were to find the limit as x approaches negative Pause this video and see, first of all, does this theorem apply? And if it does apply, what is this limit? So the first thing we need to see is does this theorem apply? So first of all, if we Out what is the limit as x approaches negative three of f of g of x. Left-hand side is our function f and what we see on the right-hand Represented right over here, let me make sure I haveĮnough space for them and what we see on the So here I have two functions, that are graphically So let's look at some examples and see if we can apply this idea or see if we can't apply it. So the limit as x approachesĪ of g of x needs to exist, so that needs to existĪnd then on top of that, the function f needs to beĬontinuous at this point and f continuous at L. ![]() Under certain circumstances, this is going to beĮqual to f of the limit, the limit as x approaches a of g of x and what are thoseĬircumstances you are asking? Well, this is going to be true if and only if two things are true, first of all, this limit needs to exist. The limit as x approaches a, of f of g of x and we're going to see Gonna think about the case where we're trying to find Of composite functions and in particular, we're Going to try to understand limits of compositeįunctions, or at least a way of thinking about limits Examples 2 and 3 in this video already fail to meet the extra condition, but they also fail to meet earlier conditions and so we don't get to see the continuity condition in action. lim (x->L) h(x) must exist, but be different from h(L). So, two recommendations: (a) Add the composition law to the "Limit properties" video, including the requirement for g(x) to be continuous at L and (b) add to this video an example where the only thing stopping us from being able to evaluate the limit is that h(x) is not continuous at L. Instead, students are left to stumble over this condition by answering some of the following exercises wrong and then trying to figure out why they were wrong. Nor do any of these examples bring up that extra condition. There's a confusing omission here in the video series, because (1) the "composition law" is assumed here but was never presented ("We can leverage our limit properties," the video says, but that property was left out) and (2) the extra condition for the composition law, namely that g(x) must be continuous at L, isn't mentioned. ![]()
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