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Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to 4}\frac{x^2 + 3x}{x^2 - x - 12} $

Does not exist

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Okay, we want to find a limit of dysfunction as X approaches for now, when X equals for you would get four square at 16 minus four minus 12, which would end up being zero. So when X is four, the denominator here is zero and you can't divide by zero. But that's not a problem because when we take the limit of X approaching for X never actually gets to equal for X, just approaches for it's not going to equal for All right. Uh So what we're going to do to evaluate this limit? Um We're going to factor the numerator and we're going to factor the denominator and then see if we can cancel some common factors. So we are going to be taking the limit as X approaches for now looking at the numerator, X squared plus three X. Let's factor out the greatest common factor. Which would be X. So the numerator can be rewritten as X times X plus straight. If you distribute this multiplication, you can see X times X is X squared plus and then X times three is three X. All right, so that's going to get put over. Uh Well, X squared minus x minus 12. But we want to factor X squared minus x minus 12, X squared minus x minus 12. Well, factor Into X -4 times X plus. Straight. All right. So, we want to double check this. Uh you can use spoil x square there it is. Outer is three X. Inner is negative four X. They combined to give you a negative one X. There it is last term times last term negative four times three is negative 12. So X squared minus x minus 12 factors into x minus four times X plus straight. No, The X-plus three factors will cancel. Um we can cancel this X plus three and the denominator with the X plus three in the numerator. As long as X plus three uh is never equal to zero because you can't have a zero in the denominator, well, X plus three would equal zero when X is negative three, X is approaching four so we don't have to worry about X being negative three. So we cancel out these X plus three factors. So as X approaches for as X approaches for this, X term is going to approach for But this X -4 term Is going to approach zero. If X is approaching for an X -4 approaches for -4, which is zero. No, As X is approaching four from the right side of four, Meaning it's gonna be a little bit bigger than for X -4 is going to be a small positive number. For example, if x is 4.14 point one minus four is 40.1, So you would be dividing by .1. When you divide by a tiny number, you get a big number. So if you're dividing by a tiny positive number, you're going to get a large positive number. So as X approaches for when X approaches For from the right side of four from the positive side we say uh this limit is going to let's write it as X approaches for from the positive side, uh X Over X -4 is going to approach positive infinity. This is called the right hand limit. His ex is approaching four from the positive side. Uh This expression is going to approach positive infinity. Here's why as extras approaching for uh this X will approach for as extras approaching four from the positive side, the right side of four, meaning it's a little bit bigger than for something a little bit bigger than for -4 is still going to be positive. So for example, if once again, if X was like 4.1, -4 would be .1. And when we divide by .1 uh we're going to get large number. When you divide by a tiny number you get a large number dividing by .1 Is like times it by 10, dividing by .001 is like times and by 1000. So four divided by a tiny tiny positive number is going to be a very large positive number. So as X gets closer and closer to four from the right side of four, uh this expression is going to get larger and larger towards positive infinity. Now, however effects approaches for from the left side Of four, we call it from the negative side. For example, if X is approaching four from the left side, maybe X's 3.99 the next minus 43.99 minus four is negative 40.1. So then you got four being divided by negative 40.1, dividing by a tiny negative number, gives you a very large negative number. So as X approaches for from the left side, in this case We're going to have X over X -4 approaches negative infinity. So as we get closer to four from the left side of four, uh this expression is going to uh tend towards negative infinity, very large negative number. So when all said and done, this limit does not exist because as we approach four from the positive side, this expression goes to positive infinity. As we approach for from the left negative side, the expression goes towards negative infinity. So the limit does not exist. There is no number that this expression gets closer and closer to and also the right hand limit and the left hand limits do not agree. So this limit does not exist