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Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why.$$lim _{x ightarrow-2^{+}} frac{x^{2}+8}{x^{2}+3 x+2}$$...

Question

Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why.$$lim _{x ightarrow-2^{+}} frac{x^{2}+8}{x^{2}+3 x+2}$$

Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why. $$ lim _{x ightarrow-2^{+}} frac{x^{2}+8}{x^{2}+3 x+2} $$



Answers

Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why. $$ \lim _{x \rightarrow-2^{+}} \frac{x^{2}+8}{x^{2}+3 x+2} $$

Okay, so we're asked to find the limit as X approaches. Negative t of X plus two over X cubed plus eight. Okay, So noticed that if I go ahead and plug in negative too, I get negative to you. Plus two over. Negative eight plus eight, which is the indeterminate form 0/0. Okay. Which means we have to do something else. Okay, so, um, we're gonna leave the top as explicit e and noticed that the numerator is a difference of is a sum of cubes. Sorry. Some of cubes X cubed. Plus too cute. Right. So we can use the sum of cubes formula to write this as, uh, factor into X plus two times X squared, minus two X plus two. I mean, four. Sorry. Plus four. Okay. So we can factor, Uh, the terms. Ex plus two from inside the limit. So this is the limit. Has X approaches? Negative, too. Of one over X squared, minus two X plus four. Okay, so now if we do direct substitution, we have one over four plus four plus four, which is 1/12

So you have really made a sex goes to It's a square with wax my state overexert Square 16 My stuff. So we cannot just pointing, because if we do that here on bottom, you get two square. That's for Linus minus two, but he's a zero, so that's pretty muddy. So hopefully we can about there. Daughter. No, thanks. Planets too. Love. Also from talk. So here to get that will you Obviously long necks and then together to there should be a one. So we do the proper minus two eggs sex. So that's minus six. So he's one of those fuckers. They explain us too. Them six puzzle. We're here orderto well, so here we have a next square. So Michelle on X and then or hear when she get out. Mind those eight so or dams Minor. Still, you exist when I say, Let's see things with get rightto you know, for X My nose to X. So that's it right there. So yes, it doesn't. I told you that. So his name it is going to be called Tau, not lead a sex goes toe to then they problematic term. I will be still does. So we're left with this expression. Thanks. That's four over X. That's one. So a sex goes through too. Well, there is no problem. Here it goes. The denominator is not zero. So just answer. Teo. That's 46 six over two. Plus what? That's sleep. Well, six. He's three. Still. So that is gonna cancel this three three. Cool. Um, so then we left, but it is two.

For this problem. We're trying to find the limit as exit purchase Negative two of X cubed minus two X square plus four x plus eight. And since this is a polynomial in terms of X that we're trying to find the limit off, we can just plug in the value that X is approaching and evaluate the limit directly. We can do that with any polynomial because polynomial is don't have any discontinuities, so we just plug in negative, too. So this is equal to negative to cubed minus two negative to square, plus four times negative two plus eight, which is negative. Eight minus two times four minus eight plus eight thes eights. Cancel. So we have negative eight minus eight, which is negative. 16 and that would be our answer FIS limit.

Limit eggs too. Negative too. Uh, experts two over X cute clothes. Eight. You pretty much have the same thing here. The other time was minus. This time it's plus, but it's the same concept, right? If it put negative to hear that you're gonna have numerator to denominator too, Right? So you have to do some factoring, right? So, you know, I said, if you have a space cubed minus be cute, it was a minus B and then a squared plus B a B plus B squared, right? What if this time it is a is a cube plus e cube? Well, still is anything but just I'm gonna change signs. It's a plus B here. Then this one is gonna be a squared minus a b close bees way. So notice the difference. This changes sign. And this changes side from this genius up, right? That is the only thing that is happening. So and this other case, we're gonna use the second world wreckers. It's plus, So this is gonna be you know, this is too right. This is too to the cubic power. So this is gonna be exposed to an over experts to right And this is gonna be, uh, X were minus, uh, two X plus four. That is, if I use this one to do the expansion once I do that in me immediately, just one, cancels, despite so that I have is the limit. X approaches negative to the top one is gonna be one right? And one over X squared, minus two X plus four. So it means that we're right. See? Negative, too. Whenever I see eggs as you put negative to write. So doesn't have one. Or negative two squared. Right? Minus two times Negative. Two plus four. This is for Mrs Ford, Mrs Ford. So we're gonna be 12. Raise 1/12 that is.


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