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...

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) = \dfrac{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.

All right, We're using a graphing utility for this problem. So here's my graphing calculator and we go into the y equals menu and we type in one plus one over X to the X power for one function and y equals E for the other function. And I played around with the window dimensions, and I decided to look at a window that goes from 0 to 20 on the X axis and negative 1 to 4 on the Y axis. And here's what the graphs look like. So the blue one represents why one and then the red one represents y equals E. So what we can see is happening is as why increases whereas the value of X increases why one the one plus one over X to the X power is getting closer and closer and closer to E. In fact, this leads us to one of our definitions of e, which is that e equals the limit of this function as X goes to infinity

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.