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Problem 66 Hard Difficulty

(a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5.

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Video Transcript

This is problem number sixty six of the Stuart Calculus eighth Edition section two point five Part a Proof theorem Foreign Part three Part be proof there and four part five. So we're going to be proving Part three and four five of Steering for and there are four states. If f n g are continuous at a and sea is a constant, then the following are also continuous at A and Part three says the functions he see constant tempts the function. F is continuous Eddie and part five of their in four states that after our by Jean if she is f d a is not equal to zero is continuous at eight. You should also recall our definition of continuity, which is the limit and sexy purchase a Ah, the function f is equal to ever be on DSO And so we approach the first part. Let's work with part A For now, Number three function CF. So we want to prove that this is continuous. Well, we start with the definition of continuity for a function f. Since the problem are since the serum gives that f is continuous, then this is definitely true. We go ahead and multiply the left and the right side by a constant C. And we are left with this for now, and we're almost there. We need it to use a limit, Lana, that allows us to bring this constant into the limit. And that is definitely the case. This is absolutely possible. It's limit function is linear so we can bring the sea value into the limit. And what we end up having here now is that this last statement shows and proves that by the definition of continuity, this new function and in constant war as long as that function wasn't initially continuous at a is also continuous at a So we have a go on ahead and confirmed part three of hearing for, and we're gonna go ahead and work on party for the fifth part of tearing for, um again, we assume we know both the limit or the function f n g or continuous, so they have this definition applicable for them. So if we consider the limit exact as expert city of this new function f divided by Jean, well, we use our limit lines again, which allows us to separate the limit limiters as expression. A f over the limit as expert is a gene. Ah, And since we know that they're both continuous, we have f a here in the numerator g iv e in the denominator And as long as Jehovah, as it says in this part of the room for as long as gov is not zero. And this is definitely true notice that we have Ah, essentially, what is f over g evaluated, eh? And so this beginning part here and that's final conclusion shows that this new function half over Jean is also continuous provided that f ngor independently continuous Addie Ah, and we have proven part five as well of caring for him.