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Find the limit; ifit exists. Identify the rule; theorem or method used; e-1 lin Ax-x+6...

Question

Find the limit; ifit exists. Identify the rule; theorem or method used; e-1 lin Ax-x+6

Find the limit; ifit exists. Identify the rule; theorem or method used; e-1 lin Ax-x+6



Answers

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} (e^{-2x}\cos x) $

Okay if we plugged infinity in to the denominator we've had a sign of one over infinity. One over infinity gets closer to zero. The bigger infinity gets and the sign of zero is zero. So we have an infinite form with a zero denominator. So let's use like petals role before we use like petals wrong. I'm gonna rewrite one over X. S. X. To the negative one. It will just help with doing derivatives. So now we're gonna use low p. Tall derivative of each willpower is itself times the derivative of the exponents derivative of extra negative one is negative one X. The negative too. And denominated derivative sign is co sine of the angle times the derivative of the angle which would be negative one X. To the negative too. Uh huh. Uh huh. Now these will cancel. And now we will have E. To the next to the negative one. Power or eat to the one over X. Power over the co sign X. -1. Which would be then co sign of one over X. And uh let's plug in infinity. E 21 over infinity, And then co sign of one over infinity. So one over infinity gets closer to zero so each the.

All right, we want to find this limit or show that it doesn't exist. Um, a lot of people think about using local tiles rule or thinking about and behavior of functions to solve these. Um, I don't like those. I think Loki tells rule, well limits have to exist before derivatives. So this is a more fundamental question that and then end behavior can be hand wavy and or confusing. So um, these are both growing, this is really big, probably really big negatively. This one's really big. Um, so let's just stop them from growing. That's going to be the least confusing. So I'm going to divide top and bottom by E to the X. So this is one over E to the X minus italy X divided by either the X one and then here we have one over either the X plus two, you do the X divided by either the access to and now this theorem I'm going to write down is one over eat the X is going to go to zero. Why? Because X is really big. Even the X is really big. Let's just have this sketch of the X in the back of our minds. You know, the X is even bigger and then one divided by really, really huge number as the number gets huger is going to go toward zero. So now we can just use that there we have the top That goes to 0 -1. So the top goes to -1. The bottom goes to 0-plus two is to, So we get a limit of negative 1/2. I think that's the most straightforward way of doing this kind of problem.

Hello. We have to find the limited expense to one and so from its mother's fun. We can see that the limit the function X upon X men as one, is not defined and a little different and expressed to one because I had exposed to find the value of the function is in 30. So we can see that the limit does no and this whole operation and.

In this question. We need to find out, limit off and then sigh one of n plus Atlanta and an n goes to infinity here. Not this time. We can rerun this limit here. And you used the lock. Property can combine that. You look into one. They would have the sign one of and, um and Nablus And here temps in here. And now we see we can push the limit inside the land. Never Ganda and outside. And so I will have the limit. And yes, the infinity. Here we have the sign one of and we can reverse damage the one of and here the reason why now? Because I can set the vehicle to one of the end and we see as and goes to infinity. We have the biggest U zero. Therefore, we can revert this limit in doing in terms under teh with you zero then psych a over pay. It is a well known limit. And then we could you one ever begin that land in the one and echoed you zero here


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