 So a picture is a face that can launch a thousand ships, or something like that. Anyway, in mathematics, when we talk about pictures, we're really talking about graphs. So let's see what we can do about graphing with asymptotes. So remember that the graph of an equation consists of all points x, y that satisfy the equation. Suppose we do all of this work, and we find that y equals k is a horizontal asymptote, and x equals h is a vertical asymptote. Remember this means that as x goes to plus or minus infinity, y goes to k, and as y goes to plus or minus infinity, x goes to h. What this means is that if we look at the points on the graph, the points to the far, far, far, far, far, far, far right or left, x going to plus or minus infinity, will have y values close to k. And likewise, the points high, high, high, up or way, way down, y going to plus or minus infinity, will have x values close to h. And something happens near the origin as well, but we might not care about it. To use a phrase that I personally detest, we might only care about the big picture. And that's really what asymptotes are all about. Because we're looking at things that happen as x goes to plus or minus infinity, or as y goes to plus or minus infinity, this is the ultimate big picture. So let's use our end behavior to sketch the graph of y equals 1 over x minus 3. And that means we'll want to find the horizontal and vertical asymptotes. So let's find the horizontal asymptotes first. For 1 over x minus 3, as x goes to infinity, we can think about x as a large positive number. Then x minus 3 will be a large positive number. So 1 over x minus 3 will be a small positive number. And so we say that 1 over x minus 3 goes to 0. Equals means replaceable, so we can say that y goes to 0. Now the correct statement of the asymptote is y equals 0. The equation of the asymptote requires a specific real number. But when graphing, it helps to have a little bit more detail. Remember that 1 over x minus 3 will be a small positive number. And we might represent this using our superscript notation, 0 plus. Again, equals means replaceable, so that means that y is going to 0 plus. And so we might consider the following for the point x, y on the graph, as x goes to infinity, y goes to a number slightly more than 0. So x, y will be very far to the right and slightly above the x-axis. And so we might draw that portion of the graph like this. First, let's draw our horizontal asymptote, y equals 0. Since the asymptote is not actually part of the graph, we'll use dashed lines to indicate it. Then we'll go to the far, far, far, far right, but stay slightly above the x-axis. Likewise, as x goes to minus infinity, x will be a large negative number. x minus 3 will be a large negative number. 1 over x minus 3 will be a small negative number. That's to say a number slightly less than 0. As an asymptote, we don't care about slightly more or slightly less, so 1 over x minus 3 will still go to 0. And so 0 is an asymptote whether x goes to plus or minus infinity. But again, it's helpful to think about this asymptote as indicating that y goes to 0 slightly less, so that for a point x, y on the graph, as x goes to minus infinity, y goes to a number slightly less than 0, so the point x, y will be very far to the left and slightly below the x-axis. So again, as we go to the far, far, far, far, far, far left, we should get close to the x-axis but stay slightly below it. Now we can't reduce 1 over x minus 3 any further, so that means we'll have a vertical asymptote at x equals 3. And let's draw this vertical asymptote in. Since this is not part of the graph, we'll use a set of dashed lines. And we'll see what happens as x gets close to 3 from below and x gets close to 3 from above. So as x gets close to 3 from below, x is a number slightly less than 3, then x minus 3 will be slightly less than 0. So 1 over x minus 3 will go to minus infinity, and that's because we're dividing 1 by a small negative number. Equals means replaceable, so as x gets close to 3 from below, y goes to minus infinity. As x gets close to 3 but stays slightly more, x minus 3 will be slightly more than 0, and 1 over x minus 3 will go to positive infinity. Again, that's because we're dividing 1 by a small positive number, and we should get a large positive number as the result. And again, equals means replaceable, and if it's not written down, it didn't happen. As x gets close to 3 from above, y goes to infinity. So for points x, y on the graph, as x gets close to 3 from below, y goes to minus infinity, so if x, y is very far down, x is slightly less than 3. So if we follow our points on the graph, as x gets close to 3 but stays slightly less than 3, our points go way, way, way, way down. Also, as x gets close to 3 from above, y goes to positive infinity, so if x, y is very far up, x is slightly more than 3. And if we follow the points on the graph, as x gets close to 3 but stays a little bit more than 3, our y values go to infinity. We go way, way, way, way, way, way up.