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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

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

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this is problem number twenty Out Stewart eighth Edition, section two point two Guess the value of the limit if it exists by evaluating the function at the given numbers, correct a six to small places the limit as X approaches negative three of the function of X squared minus three x, divided by the quantity X squared minus nine. And the given numbers are the following x values that are listed at the bottom in order to guest limit in order to I'm trying to figure out what does them it might equal. We first use dysfunction, which we will call it off and evaluated at all of these given excise. So take the first one first X value and plug it into the function and then see what the value is and continue with the following values in the spreadsheet. We have done that and we have done the calculations, and we can confirm that when you plug in next is make this two point five you get negative. Five. What next is negative? Two point nine you get eight to twenty nine. Next is negative. Two point nine five you may get a fifty nine one x a negative two point eight nine. You get negative to nine nine when X is negative. Two point nine nine nine You get made of two thousand nine hundred ninety nine and when X is negative two point nine nine nine nine you get negative twenty nine thousand nine hundred ninety nine. We will be stopping here for now because we sense a patter and we see that the closer that this value gets too negative three, which is what the limit is to find out. We see that the function approach is very, very large negative number. And since this is the one that is closest, Teo Negative three, we will just keep our eye on that. For now, we look at the other side. So when we approach negative three coming from native two point five two point made of two point nine, we're approaching it from the right hand side. The next values approach negative three from the left hand side and we see that the values of the function increase from seven to thirty one to sixty one two, three hundred won two two three thousand won and finally to thirty thousand and one. And since This last point is the closest to the value of negative three. From the left hand side, we will highlight that. And what we do is we compare because we have now been able to describe the behavior of the function around X equals negative three. What we see is that to the right of negative three, the function is has is evaluated in a very large negative number and closer to negative three from the left hand side, the value or the behavior of the function, it is very, very large, very positive. And in order for this limit to have a confirmed value, we first need to have these two limits exist. In this case they do. But then we also need them to be equal to each other. And we see that that won't be the case because one is very large thinking, remember, And the other one is a very large positive number. Eventually, what we will see, especially if we put this function, is that from the right hand signed, the function approaches Negative infinity. That's the limit from the right hand side from the left hand side at the function approaches positive infinity. And because the is too, I don't necessarily agree or equal. We say that this limit does not Chris

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