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