 Given a graph, we can also determine, or at least approximate, the values of many limits. And this speaks to a more general rule. It's useful to be able to switch between algebraic and the geometric. Algebra is about numbers and arithmetic. It leads to fungions. f of x equals x squared minus cosine 3x plus e to the 5x, and similar statements. We have formulas, we have numbers, we have arithmetic, and these are all algebraic ideas. In contrast, geometry is about position and shapes, and it leads to graphs. Limit is an algebraic concept, and in fact most of modern mathematics is built around algebra. However, it's often convenient to be able to switch back and forth between the two disciplines. And in this case, we want to find a geometric concept that is equivalent to our limit concept. And we might approach this as follows. If the limit as x approaches a of f of x is some value l, then on the graph of y equals f of x, the y-coordinates, which are the f of x values, approach l as the x-coordinates approach a. For example, suppose we have the graph of a function. So let's find a couple of different things. Let's find a function value as well as a couple of limits, both one-sided and unqualified. So let's say I want to find f of 2. Since this is a graph of y equals f of x, then f of 2 will be the y-coordinate when x is equal to 2. So I look on the graph and I see that the y-coordinate when x is equal to 2 is going to be 5. What about the limit as x approaches 2 of f of x? Since y is equal to f of x, then the limit as x approaches 2 of f of x is equal to the limit as x approaches 2 of y. So we want to find what happens to the y-coordinates as x gets close to 2. So let's start with a point on the graph and move towards x equals 2. We should also start with a point that is beyond 2 and move backwards towards 2. Based on these observations, we can determine what happens to y as x gets close to 2. And we see that as long as we're close to 2, our y-coordinates are close to 10. And so we might conclude that the limit as x approaches 2 of f of x is 10. The limit as x approaches 3 from below will be the y-values of the points on the graph as x gets close to 3, but staying less than 3. Now 3 from below and staying less than 3 is an algebraic concept corresponding to the geometric concept of to the left of x equals 3. So let's see what happens when we approach 3, but stay to the left. As we get close to x equals 3, but stay to the left of x equals 3, we see our y-values get close to 20. And this suggests that the limit as x approaches 3 from below of f of x will be equal to 20. Similarly, the limit as x approaches 3 from above of f of x seeks to find the y-coordinates as x gets close to 3, but stays to the right of it. So if we follow along on our graph, as x gets close to 3, but stays to the right of it, our y-values appear to be getting close to about 15. And so this suggests that our limit as x approaches 3 from above of f of x will be equal to 15. Finally, remember that in order for the unqualified limit to exist, we need the limit as x approaches 3 from below and the limit as x approaches 3 from above to be equal. However, this doesn't happen, so the limit does not exist. Or let's take a look at another situation. Again, we have another graph, and we want to find a bunch of limits, and this time including limits as x goes to infinity. So the limit as x approaches 1 from below of g of x, well that's going to correspond to the geometric picture of what happens to our y-values as x gets close to 1 from the left. As we approach 1 from the left, our y-coordinates seem to get larger and larger without limit. And so we might conclude that our limit as x approaches 1 from below of g of x is positive infinity. On the other hand, as we approach 1 from the right, our y-coordinates seem to get more and more negative without limit. And so we might conclude that the limit as x approaches 1 from above is negative infinity. How about this concept of x going to infinity? Well, this is an algebraic concept corresponding to the geometric concept x is getting farther and farther to the right. And we see that as we go farther and farther to the right, y seems to get closer and closer to 1. So we might conclude that our limit as x goes to infinity of f of x will be 1. And finally, this concept as x goes to minus infinity corresponds to the geometric concept of getting farther and farther to the left. And as we go farther and farther to the left, y seems to get closer and closer to 2. And so we might say that the limit as x approaches minus infinity of g of x is going to be equal to 2.