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Problem 15 Medium Difficulty

Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit and illustrate with a diagram like Figure 9.

$ \displaystyle \lim_{x \to 3}(1 + \frac{1}{3}x) = 2 $

Answer

$\delta=3 \varepsilon$

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Video Transcript

Okay, they want you to do an excellent delta proof to show that this limit here is equal to two. So let's look at a picture and see what's going on. First of all, the function F of X is one plus one third X. Which is a line Who's y intercept is Juan and its slope is 1/3. So up one third over one or up 1/3. So up 1/3. Okay, so here's the line. Yeah, yeah, here it comes again. Going to try uh here guys. Yeah. Okay. So what's happening in an epsilon delta proof is you're gonna tell me how close to two you want and then I'm gonna tell you, oops, yeah, How close to three you have to be on the Y axis. Uh That's a terrible picture. But okay, so you tell me how close to two you want to be. I will tell you how close you have to be 23 down here. Okay, so that's what's going on. So here comes the proof. Yeah. Okay. Okay. So here's what we need to do. We need to show that for all Absoluteal greater than zero. There exist, wow, so messy writing. Let me fix that. Okay. Mhm. We need to show that for all. Excellent. Greater than zero. There exists uh delta greater than zero. Such that the distance between the function and the limit. That's the absolute barry. That's the distance between them is less than epsilon. When or whenever The distance between X and three which is coming from right here. This last thing, delta. Okay, so what is really saying is you tell me Absalon, I'll find delta for it. You tell me how close you want to be 22 on the y axis, I'll tell you how close to three you need to be on the x axis. Okay, so then what you have to do is you got to do a little bit of work on the side here. So this is not in the proof, this is me doing calculations. So start with the ffx minus l. So as the value of one plus one third x minus two and said it less than epsilon. So that's 1/3 X -1 is less than ε. So what you're trying to get to is X -3? Oh so I'm gonna just factor out a 1/3 here. Okay, convince yourself that when I factored out of 1/3 it's really X -3 And now I'm going to multiply both sides by three, so I get X -3 is less than three. Absalon Okay, so that is what I'm gonna choose delta to be. So if you tell me you want within one of two on the y axis, I will tell you you have to be within three of three on the x axis, you want to be within one half on each side, you have to be within three hats on on the X axis, you want to be within 1 100? You have to be within 31 hundreds of three here. Okay so it's like a magic formula. Okay, so then here's what you say next. Okay, remember this stuff in the in the squiggly is not in the proof You do it somewhere else. Okay. So you say get a pen. Okay. And then you say let Delta equal to 3 ε then one plus one third X minus two is what is equal to 1/3 x -1 Which is equal to 1/3 X minus one. Which is less than oh explain history, sorry, Which is less than 1/3. Okay, these are all equal that's really equal to this, that's really equal to this. But this is actually less than one third delta. But we've chosen debt, we've chosen delta to equal to three. Absalon. So this is equal to one third times three epsilon which is equal to obsess on. So here's what we just showed. Okay, look at the screen. F index minus L is less than epsilon Whenever X -3 is less than Delta. Okay, now we're done. So since one plus one third x -2 is less than ε? Wind. Oh, I don't know what's wrong with my best sense on there. Oh, I forgot. The salon. Okay. All right. So since this is less than epsilon. Whenever X -3 is less than delta then limit As X approaches three of one plus one third X really does equal to two. Okay, so you're right. Everything in the black open this last part, which is accidentally and read what's oh my gosh, okay. So not that stuff in the flowery cloud, but all the way to here. I hope that helps. It takes a while and then all of a sudden one day you're gonna wake up and you're gonna say, Oh, I'm just saying how close to three I have to be to get close to two. Okay. Hopefully that will happen.