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Prove, using Definition 9, that $ \displaystyle \lim_{x \to \infty} x^3 = \infty $.

$\lim _{x \rightarrow \infty} e^{x}=\infty$

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University of Nottingham

this problem Number seventy eight of this tour Calculus C edition section two point six prove using definition nine that the limit is expertise. Infinity of X cubed is equal to infinity. We're going to call definition nine. That states that for for every that, if X is greater them in this will guarantee that the function is greater than the value em. So we confined any value. Ah Mm. And that satisfies this for value em. As long as we can consistently prove that this is this, this will be true then this infinite limit will definitely I'll be true. Okay, So what our next step is to do is to try to confirm this. Let's start with this second part here. The function X cubed greater than M and what this leads us to believe is part probably that X greater than Cuba to them would also be a good restriction. I know comparison between ex greater than an X is greater than this, so it may be appropriate from this. We're always here to choose an M that's equal to end cubed. So we're going to try this out on DH, see if there's something that is consistent for with this choice. So we'LL begin with his first statement X cubed critter than Emma. And again we're just going to choose him as and cute and then see how this works. Let's go subject and cute to both sides, and this is a difference of cubes, so that could be factored as explains and X squared plus x times them was in squared, created in zero. And then we just consider what each of these terms are equal to. Is this term greater than zero? Well, our X values in this case as we approach infinity, are always positive We're restricting our ex Don't mean to be just approaching positive affinities with each of these guys X squared and X times and are potentially positive. And I value that we confirmed to be positive as well. So and and on scored is always positive, regardless but and times X, we confirmed, since we restrict them to be positive and acts to be positive, that that is also a positive value, so this term will be always positive. So it's just a matter of making sure that this term is positive. So we ask ourselves, How do we make sure that X minus n. It's positive. All we have to make sure just that X is greater than that. And this is our proof we have reached based on the choice of them is equal to an cubed. We have reached the other condition and it's being met, or the results of this assumption is that X is greater than end. So this proves our limit based on the finish. Nine. That if X is greater than him and the question will definitely be greater than and which we chose and in Cuba in this case. But overall, we have proven the limit based on definition night.