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Problem 42 Hard Difficulty

Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to -6}\frac{2x + 12}{| x + 6 |} $

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okay we want to find the limit as X approaches negative six if possible of this function. Two X plus 12 over the absolute value of X plus six. First thing I want you didn't notice is uh let's go ahead and take the limit. His ex approaches -6. I want to factor out a two out of this numerator. Um If we factor out at two out of two X plus 12 we will get two times in X plus six. And I wanted to do that because that X plus six factor that I have here ah Relates to the absolute value of X-plus six that I have in the denominator. So uh things are going to be interesting when we look at this limit as X approaches negative six because we're really going to focus we do have this two here but we're really going to focus on X plus six over the absolute value of X plus six. Uh It may be possible that X plus six is positive. Of course absolute value of X plus six will always be positive because absolute values always positive. So you might have a positive over a positive like you know maybe a total of 5/5 which would be one But it's also possible that X-plus six might be negative. Um where is the absolute value of X-plus six would still be positive. So let's say for example if X-plus six was equal to negative 10 then the absolute value of negative 10 would be of course positive 10. So you would have a negative 10 over a positive 10 which would equal negative one. Um so that's what's going to happen as we that's what we're going to look at. As we look at the limit of this expression as X approaches -6. Now for a limit to exist, it has to exist from both sides. Yeah, so we're going to look at the limit of this expression Okay, this one right here as X approaches negative six from the positive side From the right side of -6. But we're also going to look at the left side limit Limit as X approaches -6 But from the left side or from the negative side. So the limit of this expression as X approaches -6 from the left side. The negative side. No, Let's look at this over here. 1st Effects is approaching negative six from the positive side. Think of your number line. Okay, lets zero B over there. Uh Let negative six B over here. Uh negative five will be over here. Um Let's make this a little bit longer. Okay, let's squeeze in a -7. Okay, as X is approaching negative six from the positive side, that means X is approaching negative six. Negative six is right here. We're approaching negative six from the positive side. So for approaching negative six from the positive side, maybe we're at something like negative 5.9. Well, if X is negative 5.9, negative 5.9 plus six is going to be positive 0.1. Basically the most important thing I want you to know is that as we approach six from the right side from the positive side as we approach negative six from the positive side, X-plus six is going to be positive. And so X plus six which is positive over the absolute value of X plus six, which will be the same amount and also positive A positive X plus six over the absolute value of X plus six is going to equal one. So this entire limit here approaches did number two because a positive X plus six divided by an absolute value of X plus six is gonna be one. This is a positive number and this is the same positive number. So the same number over the same number is one. So two times one is two. That's why the limit of this expression as X approaches -6 on the positive side is too No, Let's look what happens to the limit of this expression as X approaches -6 from the left side from the negative side. If we're approaching negative six from the left side from the negative side, Uh that means we might be over here at something like negative 6.1 Effects is negative 6.1, negative 6.1 plus six is negative 60.1. The most important thing I want you to take away from this is if we're approaching negative six from the left side for approaching negative six from the left side then X plus six is going to be a negative number. For example, X could be negative 6.1 negative 6.1 plus six is negative 60.1. So if we're approaching negative six from the left side, X plus six is going to be negative but the absolute value of X plus six is going to be positive. So you might have something like negative 60.1 divided by positive 0.1 which is negative one. Ok. A negative 10.1 divided by a positive 0.1 is negative one. So what you really have here is to Time Zing -1 Which is -2. And so we see uh that the left hand uh and right hand limits as we approach negative six are not the same. Okay, the limit of uh this function as we approach negative six from the positive side is too But the limit of this function as we approach negative six from the negative side was negative too. So the limit of this function as we approach negative six does not exist because the right hand limits and the left hand limits do not agree. They are different values. So this limit does not exist