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Problem 44 Hard Difficulty

Suppose that $ \displaystyle \lim_{x \to a} f(x) = \infty $ and $ \displaystyle \lim_{x \to a} g(x) = c $, where $ c $ is a real number. Prove each statement.
(a) $ \displaystyle \lim_{x \to a} [ f(x) + g(x) ] = \infty $
(b) $ \displaystyle \lim_{x \to a} [ f(x) g(x) ] = \infty $ if $ c > 0 $
(c) $ \displaystyle \lim_{x \to a} [ f(x) g(x) ] = -\infty $ if $ c < 0 $

Answer

(a) $\lim _{x \rightarrow a}[f(x)+g(x)]=\infty$
(b) $\lim _{x \rightarrow a}[f(x) g(x)]=\infty$
(c) $\lim _{x \rightarrow a}[f(x) \cdot g(x)]=-\infty$

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Video Transcript

This is Problem forty four of the Stuart Calculus East Edition, section two point four. Suppose that the limit is expert Tuesday F is equal to Infinity and the Limited's experts. A gene is equal to C or C is a real number. Prove each team it. And for each of these statements, we'LL take them one by one and we'LL use this information here for F on this information. Yeah, since they're f is an infinite limit we use and as the determining value that for any end there exists a delta that makes these inequalities true that the function is greater than in ah, for any Delta. And then for Jean. We used to tell the absolute definition that the absolute value of the difference between G and seeing his lesson, it's lan for any Delta that satisfies these conditions. So we'LL begin with party up here party. We have a new limit. There were supposed to prove, and it's a limit. His experts say that the function of plus Jean and that has the equal to infinity since it's an infinite limit, we have tio engage the proof based on a new value for this infinite limit for affray reason and erinys em. And we're going to say that we need a delta greater than through, um such then, if the absolutely of the difference between X and A It's less than this Tarlton. They weren't going to have this new function. Plus, Jean the absurdity of this. Your function Sorry, not think about it. You just any function? All right? I'm being greater than this. M value. So what we're going to do in order for us to add these two values after Gene, we need to first figure out what a good G value would be if we see here the criteria for jeans at the observatory of the different agencies. Listen up, Slim, which is the same as thinking of Absalon less then the different contingency lesson. Absalon, I'm still here. In this first part of inequality, we can actually see that gene if we had seat aged. Her G is greater than C minus Absalon. So we know we know that here its creator then C minus epsilon. And we know that efforts created an end. So top us out. We can write to that at the function f sixteen. It's greater than or can you replaced with in And Seaman is he and scene that's you. So to some of those two is creator than that's some, right, because each other functions are greater than those terms. And we want to make sure that this is created that m as we saw here, Tom. And so we will take on some values from assumptions for and see. But on this he is actually Absalom. Just correct that real quick. And we're going to do this for three different cases. Since C is not specified in party, I'm going to do it for the case. What does CIA Ciro and then in the next one will be? What is what if she is created from D'Oh and then finally would have ceased lesson to your own came. So for AA C is equal to zero, we will choose and be equal to to him and absolutely equal to M. And we're allowed to do this because F N g. The first limits. I'm are proven and they're able to work for any choice of end or absolute. So we make these choices and then we see that we can't to m CNN's hero my name's Sam. That means that this is M. And if this is equal to em Ah, Then this statement here, the effortless cheese grater than him is true. So that is proven for Sean's equal to zero. Mercy is greater than zero. We will choose, then vehicle to em. That will be our choice for him. Uh, absolutely. We got to see. And this work of us MM plus C minus e is equal to em. And this is true. And that confirms our case. There finally, for sea is less than zero. Ah, very similar to the previous case and is equal to M. Absalon is equal to negative scene. Or actually, we're going to make it end equal to M minus to see again, out of convenience because we'll see that will have them minus two. Seeing scene minus Absalon Absalon being negatives here. So finest thing it is he is closing and then we see that and mine is to see plus two C is equal to em and we have shown that part is true. Um, created these arguments for part B. We have a new function that we're looking at. We're looking at the function F, f and G being multiplied. And it's another infinite limit. We're looking for the same conditions appear in party except that the product must be created them this valley that month we were talking about Previously on this since C is greater than zero, we're going to select salon such that Absalon is between zero unseen. And if we and some checks in on the sign I'm sorry, he and rotate we see that we get C minus Epsilon. This created material and us a choice of n equals em over C minus epsilon In the great choice, you will see why. Because the search of the product of these two effin gene I don't recall that half his greater than n and G is greater than C minus scene. So the product eyes creator than and times seem as he and with our choice of and em over Time magazine, we see that he's cancelled and we're left with him. And we have proven that this f times cheese creator, then now for part, see, and in a very similar proof, I'm except for negative infinity. For this I'm going Teo at another page. And so her and calls to have a you know, to have a function g f times change and it's going to be less than this and Valium, and that's what we do with, ah, negative infinite limits. So Haller conditions are the same. Um, we begin with condition That's he Ah, it's less than zero We'LL see being resting Tio means that I need to see is greater than zero and we'LL choose and absolute value between zero and see such that we have c plus Absalon being a lesson zero Now we'LL choose and is equal to em over people seen on which we know is created that zero and well proceeded with. Well, actually revisit this Gina, any quality Get over here G minus scene lesson Absalon And, uh, we're going to show on this right side. Since we're dealing with this person, the cases he's listens around two eyes Liston C plus We could also write This's negative teen, its creator than negative the quantities Absalom King. And if we do it in this form, we also recalled f greater than and and the reason that we put in this forms because thes to have to be greater stand there such that we can multiply the two now. Effigy multiplied is definitely greater than and times negative. C plus. I'm sorry. I'm sorry. This is negative, G. So this will be a negligee. So Deaf times negative team its creator then and times negative. See Praveen. After doing that, we can multiply it. Negative one about science giving us just after I miss Gene on the left. Now this is less than and Time C plus Epsilon. Remember that our choice for and was appear em over C plus epsilon and we see that the specific choices convenient and it cancels our surplus. Absalon. And as such, we are able to finally get it. And if the Knicks times change Rex, it's listing any value him and we first about the size. And so for partying from this has been proven so the limit from towards singing affinity eyes true and we have completed the proofs for each part a bnc