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Problem 35 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \arctan(e^x) $

Answer

$$\frac{\pi}{2}$$

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Video Transcript

All right. Um We've got a really cool limit question. We've got our tan of either the X. That's fun. Um Okay so we need to think about what tan and our town looked like. So let's just look at tan first tan of zero is zero And then between negative pi over two and pi over two we get up to and actually it doesn't go flat. The difference between tan and X cubed is whether they go flat at the charge, there's more differences but for sketching purposes. Okay, so this is um Tan X. And there's a there's a lot more of them. But we don't care because when we take the inverse function we only take one part of the domain so that the inverse is still a function. Otherwise once we took the inverse that wouldn't satisfy the vertical line test. So now we're just going to switch the X and Y axes or reflect across the line by equals X. Whichever way you prefer to think of it. But we're only gonna do this piece and then this is this is the function. Um Oh, sorry about that. Why equals arc tan X. Okay. Now, um what happens as X? It's really big? Italy excess is even bigger, right? Just think about have this graph of E to the X in the back of our minds to this is why he calls me into the X. It's very, very big. Um Okay, so we're looking for are tan of a big value of X. Arc. Tanne of a big value of X looks like that's going to be pretty close to pi over two. So, um we just solved the season grass so that equals pi over two. Yeah.