Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become...

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become large? What happens as $(x, y)$ approaches the origin?
$$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$

Right. So let's see what happens is excellent. Why become large? Here's the plot here, so you can see his ex is getting bigger. We're going toward this value of one half. Where's wise going? Bigger. We're going to the value of negative. So we're either going towards one half or native one half. And what's happening at the origin? Well, we see that at the origin here. Zero zero, we're going down. This is a deep hole just going down further and further towards negative infinity. Okay, so that's exactly what's happening.

Okay, so we're asked to use a computer to graft? Uh, yes. And, uh, we want to know what the limiting behaviors are. So let's just look at the graph and see it. But we can deal. Okay, Super. For positive X and positive y, what happens is X and y go to infinity. Positive. Once lengthy grabs goes to zero, you're free. Look. And that makes sense because the growth of X squared plus y squared is going to be larger then X plus y. So since the bottom is growing faster than the time, it will go to zero as well and appears that from negative X negative light we're also approaching zero, although from under the sea point. And, uh, you'll see that there is a split here along y equals X And what the split is is essential notice that if y is greater than next, all right in X is negative in the magnitude of Hawaiians, Creator will be in the positive. But if the magnitude of excess greeter and were negative won't be in the negative and so that really doesn't matter, though, because when we divide this by something that is much larger than it. So this numerator here, this x plus y both of them are going to go to zero. All right, so we know what happens when X and y approach positive. Uh, negative Infinity. Uh, so what about when x and y approach zero? Well, we know they can't go to zero because we have a point discontinuity, but from the Griff, it looks like the function does actually approaches euro. And if we get course waken, see that the graph does actually approach zero. What's going on next to zero is again related to the fact that the denominator has Squyres at it and the numerator as X amount into the first power. And so when X and Y are both small, their growth read in terms of health us, the shrink is faster, the next expose going on top. So you'll see that the graph sort of grows. Ah, infinitely high near on this sort of region where X and y it have a magnitude less than one. But you know, in between negative 10 in 01 So, between this region, the denominator is getting small very, very fast in the numerator is getting small at the and the slower read to grab is in the value of F is increasing. Mhm. But despite that, we can see that it does actually approved 20 As we uh huh, go to zero from pretty much every angle on the ground. All right.

We're going to graph the function F of X equals log base two of eight X on a graphing calculator and then analyze the graph. Here's my calculator and I went into Why equals typed in the log rhythmic function. And I went to a window and change some of the numbers to numbers that I thought would make sense for this graph. And you can always fiddle with these numbers. And, um, there's no one right window, but just come up with numbers that makes sense for your graph. You can check the graph and then change the window dimensions again if you need to. And then I press graph and here's what we see. So we're gonna analyze this for many of its features. So have captured a picture of the graph, and now we're gonna look at the features such as domain and range. When you look at the graph, you can see that the X values must be greater than zero. So we could say X is greater than zero. Or we could use the interval notation zero to infinity, and when you look at the graph, you can see that the range, which is the y values. They go down, they go up infinitely far down, even though sometimes your graph on your calculator looks like it stops. It's really continuing to go down further and to go up further, even though it's going up slowly. So the range is all real numbers, and we can write that as negative infinity to infinity. We also see that this is a continuous graph and it is increasing. Basically, increasing just means going up from left to right and then for its ass and tote. We know that log rhythmic graphs typically have a vertical Lassen tote, and so we can see that are vertical Assam tote. Is this line here? X equals zero and for the end behavior, let's take a look at each end and describe the end behavior with limits. So to the left exes just going toward zero. So it's to the right of zero, and the Y values are heading toward negative infinity and for the right end behavior. When X goes toward infinity, F of X is heading. Excuse me for my other limit. As X goes to zero f of X is heading to negative infinity. I left my negative off. And when X is going to positive Infinity, FX is going to positive infinity as well. So the exes go up, the wise go up as well.