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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 $ = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999

$$\text { For } f(x)=\frac{x^{2}-3 x}{x^{2}-9}$$

$$\begin{array}{|l|c|l|c|}\hline x & f(x) & x & f(x) \\\hline 3.1 & 0.508197 & 2.9 & 0.491525 \\

3.05 & 0.504132 & 2.95 & 0.495798 \\3.01 & 0.500832 & 2.99 & 0.499165 \\

3.001 & 0.500083 & 2.999 & 0.499917 \\3.0001 & 0.500008 & 2.9999 & 0.499992 \\\hline\end{array}$$

$$\text { It appears that } \lim _{x \rightarrow 3} \frac{x^{2}-3 x}{x^{2}-9}=\frac{1}{2}$$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

So in this problem were asked to guess the value of this limit if it exists by adding the function at the given numbers and were given a limit as X approaches three of the function X squared minus three X over x squared minus nine. Well let's notice something here first of all, Notice that this is the limit as X approaches three of x times X -3 over X plus three times X minus three. And so those cancel. And so we have the limit As X approaches three of X over X plus three which is 3/6, which is one half. So expect that these numbers were going to evaluate Will all approach 1/2. So now let's see what happens. So I have a value of X here and the value of my function. Okay I'm asked to do this at 3.1 3.5 three points oh one three point 001 and three point 0001. Okay so let's let's do these right, these are all approaching three from the right aren't they? Because their numbers greater than three. Okay. So if I go for a second to my graphing calculator and I put in the function um must do the parentheses X. Weird minus three X. Those are differences divided by X squared minus nine. Okay. And I look for a minute around value three. Let me fix my wife scale here for a minute. Go 0- one. Okay now Here's three right here. OK and there's a half. So you notice that the function right, goes right through that half there, doesn't it? Right through that half right there. Okay. So what I can do is look at these values 3.1 3.1 is here is here and I need It said to six decimal places. Well let's see. I could do is go into my calculator. Go 3.1 squared -3 times 3.1. Is that divided by Princess? 3.1 squared -9 goes out. My breath sees there you go. Can't be right to clear all this out. 3.1 squared minus three Times 3.1. Is that divided by Princess 3.1? 33 3.1 squared minus nine. Those princesses that Okay, So 508 196 Would be six decimal places or 197 508197. Okay, so go back over here. Back to here. 0.5. What we say? 508197 0819 197. Okay, good. 197. Okay. And if we put it in as 305, that in our calculator as well. And calculated then what we'll get is 0.5 04 132, Putting 301 in there, we'll get 0.500 eight 3 1. You know this is getting closer and closer to .5 and 0.5 0008 three. To make that a little bit neater there And then three triple on one. A 0.5, 0000 eight. Okay, then the next phase were given are 2.9, 2.95, 2.99, 2.999, two point 9999. We plug those in and calculate them like we did the others. You'll get 0.49 15, And then we'll get 0.4 95 798 And a zero 499 165. And then 0.4999 16, 0.4 999 91. So as we can see These are all approaching .5, which is a half. So this limit as x approaches three is a half.

Oklahoma State University