 It's useful to be able to work back and forth between the algebraic and the geometric. And so while we've found limits from graphs, we've found the algebraic from the geometric. It's also useful to go in the other direction and find the graphs from limits to find the geometric from the algebraic. So a key point to remember is that on the graph of y equals f of x, because y equals f of x, anything we know about f of x is also something that we know about y. And remember y is the vertical displacement of the point. In other words, on the graph of y equals f of x, y is going to tell you how far above or below the horizontal axis the point on the graph is located. For example, suppose I know some algebraic facts about f of x. Based on this, let's see if we can sketch a graph of y equals f of x over the interval between 2 and 4. Because it's the most familiar, it's probably best to start with the actual function values. Since y equals f of x, then if we know a function value, we also know the y value. So let's take a look at this. f of 2 is equal to 8. And what this tells us is that if x is equal to 2, y is equal to 8, and so the graph goes through the point 2, 8. We have another function value, f of 4 equals 5. So this also tells us a point on the graph. And in particular, it tells us that x equals 4, y equals 5 is a point on the graph. And so the graph also goes through that point. Again, since y equals f of x and the limit as x approaches 3 from below of f of x equals 5, then that says the limit as x approaches 3 from below of y is equal to 5. And we might read this as as x gets close to 3 from below, y gets close to 5. In other words, as we move right along the graph of y equals f of x, we approach the point 3, 5. There's one important caution here. This does not tell us anything about what happens at x equals 3. Remember, the value of f of x at x equals a is not relevant to the limit, and that also means the limit tells you nothing about the function value. So we don't actually know what happens at x equals 3, so we can't put a point at 3, 5. All we know is our graph is getting close to this point. Nevertheless, it's convenient to have a target here. So we'll do that by putting an open circle at the point 3, 5. And that indicates that this is not actually a point on the graph, but it's a place where the graph is moving towards. And then as x gets close to 3 from below, we have to make sure our graph approaches that point. And so our graph is going to look something like this. Similarly, the limit as x approaches 3 from above of f of x equals 7 means the limit as x approaches 3 from above of y is equal to 7. So as x gets close to 3 but stays above it, y gets close to 7. And so we approach the point 3, 7. And once again, we don't actually know whether there's a point there, so we'll draw an open circle to indicate no commitment over whether there is a point at that location. And as we move left on the graph, we should be approaching this point 3, 7. And this is what our graph should look like. Well, that was fun. Let's see what else we can do. So suppose I have some limits as x goes to infinity. Since y equals f of x, then limit as x goes to infinity of f of x equals 3 means limit as x goes to infinity of y is equal to 3. So as x goes to infinity, y goes to 3. And we can view this geometrically as follows. As we move way, way, way off to the right on the graph, our vertical coordinate gets close to 3. And this corresponds to having a horizontal asymptote of y equals 3. Likewise, the limit as x goes to minus infinity of f of x equals negative 3 says that as we go way, way, way off to the left, our vertical coordinate gets close to negative 3. And this corresponds to having a horizontal asymptote of y equals negative 3. Notice that we're also given the limit as x gets close to 5. But that limit is infinity. So it's worth remembering that when we say that an unqualified limit is some value, we also mean that the limit as we approach the value from below and the limit as we approach the value from above are the same. And so we know that the limit as x gets close to 5 of f of x being positive infinity means both that the limit as x approaches 5 from above is infinity and that the limit as x approaches 5 from below is positive infinity. Now, limit as x approaches 5 from above being positive infinity means that our y values get larger and larger as we approach x equals 5 from the right. And the limit as x approaches 5 from below of f of x being positive infinity means our y values get larger and larger as we approach x equals 5 from the left. And so this suggests that the graph has a vertical asymptote at x equals 5, where the y coordinates become unbounded as we get close to 5. And we can use this information to put together a sketch of our graph.