4

F (* _ 9)(x + 2)d....

Question

F (* _ 9)(x + 2)d.

f (* _ 9)(x + 2)d.



Answers

$$f(x)=\frac{3 x}{x^{2}+9}$$

So for this problem we're gonna have the F of X is equal to 1/9 minus expert. Before we look at the graph, let's consider algebraic lee what this is. This is the same thing as one over negative X plus three x minus three because of that. Because this is the same thing. We see that we do not want any of these values to equal zero. Because of either of these values in the parentheses equals zero. The whole thing is undefined. So based on that, we see the extra nautical three and extra nautical negative free. And we see that in the graph, that's the case too, X equals three and X equals negative three are defined. So we say that packs not equal plus uh minus three And that's going to be our final answer.

We're gonna use the table to calculate the limit of the function. Negative one over three plus X as X approaches. Negative up. And we can see why. This is why night of three is interesting, because when you plug in negative three for X, the denominator becomes a hero. So but there's a table but x here and f of X on this side. And since again the interesting behavior happens at negative three work into pick numbers that get closer and closer to negative three. What? So here these three choices are all bigger, the negative three. So that means we're calculating the limit approaching from the right. So when I plug in a negative 2.5, I get that F of X is negative, too. When I plug in negative, 2.9 f of X is negative 10. And finally, when you plug in negative 2.99 you get that f of X. It's negative 100. So we see here that a small change in X as we a closer to negative three causes a huge decrease in f of X, which tells us that the right sided limit here is negative infinity Now we're not done yet. We need to calculate the limit coming from the other direction. So the left sided limit. And they're not just putting quotation marks here, so I don't have to rewrite the whole function. And so we're gonna pick similar numbers coming in from the other direction. So let's say negative 3.5 negative. 3.1 and negative. Three point 01 So f of negative 3.5 gives us two f of native 3.1 is 10. Enough of negative. 3.1 is 100. So just as above, we see that a small a small increase in X as we get closer to negative three, cause it a huge increase in f of X, which tells us that this limit is infinity. And you can check this by graphing your function on graphing calculator or well, from Alfa. And if you do, you will see that it looks something like this. James. Sorry I got this. I got this part wrong like this, just as we would expect from our table

So the number 13 we're gonna be looking at. What happens if I approach negative three from the left or the right? We're gonna be using tables to see what happens, and that is plugging in several numbers that are closer and closer to the number. You're interested in seeing what happens to the function value. So here's my table listed out, and it's just a matter of some calculator jockeying to crunch through this. But here's what you will get. So let's see what the results are. I'm doing it 15.75 So these are all you know, rounding a bit. 150 five. 1500. So on the left, as I get closer to negative three from the left, we're going towards positive infinity and then on the right. Let's see what's happening. Negative. 14 0.25 Negative 149.25 Negative. 14. 99.25 We're going to negative infinity. So as I approached negative three from the left, we're going the positive infinity from the right. I'm going to negative infinity. Well, we only plugged in six numbers. How do we know we're going to offend thee all. Well, that's very class in tow. If I try to plug in negative three appear what happens? I get zero on the bottom in nine on the top nine, divided by zero. That's undefined, but it's showing the presence of a vertical Assam tone. There is infinite blow up. We just had to decide whether it was positive or negative, and it is positive on the left negative coming in from the wrecked.


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