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Problem 20 Medium Difficulty

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{x \to -3}\frac{x^2 - 3x}{x^2 - 9} $,
$ x $ = -2.5, -2.9, -2.95, -2.99, -2.999, -2.9999, -3.5, -3.1, -3.05, -3.01, -3.001, -3.0001

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Video Transcript

I believe this problem is best answered by using a graphing calculator. It's an easy way to evaluate your information. But if you don't have one you can always use a regular calculator. But basically we need to find the limit is if it exists by testing values that are really close to negative three, approaching from the left hand side and approaching from the right hand side. So if I was doing this by hand I would let X equal negative 2.5 and then I would plug it into my function and then I would evaluate that and that will end up being negative five. But with my calculator I can evaluate it a little faster than that. So opening up my T. I 84 I choose Y equals clear and turn off the things that are also on. And I'm going to type in this particular function that I was given. So I'm gonna use parentheses X squared minus three X. Close my parentheses divided by X squared minus nine. Now X has its own button up here. Here's the squared So now I have the function in here and I'm going to do I'm going to look at my table features. So second table set and I'm going to come down here to independent variable and I'm going to select ask. So that's going to allow me to plug in the values that I want to test. Now above graph is the word table. So second table now there's nothing in there yet. But now I'm going there will be as soon as they start typing in values. So the first one I want to test is negative 2.5 negative 2.9 negative 2.95 negative 2.99 negative hopes that was supposed to be negative 2.95 Oh, no, I did that. I apologize. So negative two point 99 nine. And the negative 2.999 nine which might calculator rounds 23 So it appears as we approach that value from the left hand side we're actually getting to really large values in negative territory. So the absolute values of all these are getting larger and larger and larger. Now I'm going to test the other numbers negative 3.5. So now we're approaching native three from the right and again my calculator has to around those two because it can't fit all those digits in. But it appears that what's happening as my graph approaches negative three from the right. I'm getting really large values. So in this case the limit does not exist because it appears that as X approaches negative three from the left, your limit is approaching negative infinity and his ex approaches negative three from the right. Your limit seems to in approaching positive infinity so it does not exist