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Problem 37 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{1 - e^x}{1 + 2e^x} $


\lim _{x \rightarrow \infty} \frac{1-e^{x}}{1+2 e^{x}}=-\frac{1}{2}

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Jacqueline L.

August 23, 2021

lim as x approaches (e^-2x times cosx)

Video Transcript

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.