No matter how hard you try to get to them, asymptotes remain out of reach. Learn about these invisible lines on graphs that show you places your equations just can't go.
Running to the Finish Line
Regardless of how large x gets, y will never reach 0
There's this old paradox that says that to get to any finish line, you first have to get halfway there, no matter where you start. If you're running a race, you first have to get halfway to the finish before you can actually get to the finish. But, when you're halfway to the finish, you have to get halfway of what's left to finish, so that you're three-quarters of the way to the finish. No matter how close you get to the finish line, you can only get halfway to the finish from where you're at. Eventually, this gets very frustrating. You have to wonder, is the finish line is just out of reach?
Let's talk a little bit about asymptotes. Like our finish line that's out of reach, these are places that you can't quite get to. For example, if we look at distance to our goal, or distance to the finish line, as a function of time, every second we get halfway to the goal. We get closer and closer to this horizontal x-axis. But we never quite get to the finish line.
Mathematically, we could write this as, say, y=1/x. y is the distance to the finish line and x is the amount of time that we've spent trying to get to the finish line. No matter how big x gets, y will never actually get to zero. No matter how long you keep going, you can't quite get to the finish line. You're always going to be just next to it. But if you're just next to it, your next step can only take you halfway to the finish line, which still isn't there.
Let's take a look at an example: y = x/(x^2 + 1) + 1. Let's graph this out.
Graph for y = x/(x^2 + 1) + 1
Let's define one of these horizonal asymptotes. If y approaches some number, like y goes to N as x goes to +/- infinity, then the line y=N is a horizontal asymptote. In the case of y=1/x, y is approaching zero as x gets really big, as x goes to + infinity. This is like saying that as time goes to infinity, the distance to the goal is going to zero, but it's not actually going to hit zero. You're not actually going to get there.
This graph looks kind of like a heartbeat. As x gets really, really large, we move to the far right side of this graph. And indeed, we can keep going far beyond this graph, as far as you can imagine. As x gets really large as we go as far as you can imagine to the right, y is going to approach the value of 1. On the other side, as x gets really, really small, as we move far to the left in x, y is also going to approach 1. Although it's going to approach from below, whereas for the positive values of x, it's going to approach 1 from above. Let's look at this right-hand side in more detail. When x is equal to 1, y is 1.5. As I increase x to 5, y slowly approaches 1; it's 1.19. y is 1.10 when x is 10. When x is 100, y is 1.01. And, at x=1,000, y is 1.001. As you can see, as x gets larger and larger, y is approaching 1. But it's never actually going to reach 1. You can't solve this equation for y=1.
With horizontal asymptotes, the key is what happens as your independent variable gets very large, either large in the positive direction or very large but negative.
As x gets closer to equaling 1 on the left side of the graph, y gets larger and larger
There is a second type of asymptote that we often see. This is kind of like if you're trying to go 1 mile from your house, but there's a big hill just before the 1-mile marker. You can try to climb up that hill, but it just keeps getting steeper and steeper, the closer you get to that 1 mile. This is what we call a vertical asymptote.
We see this in the case of y = 1/(1-x). As we get closer and closer to x=1 from the left-hand side, y gets bigger and bigger and bigger. In fact, y is not defined at x=1. If you try to plug x=1 into this equation, you end up with y=1/0, and that doesn't make any sense. Let's formally define vertical asymptotes. These are what happen if y goes to infinity or negative infinity as x approaches some number, then the line x=N is a vertical asymptote. We call it a vertical asymptote because it's a vertical line.
Let's look at an example. Let's say that y = 1/((x - 1)(x + 2)). At both x=1 and x=-2, y is undefined. But what happens as we get really close to those values? As we get very close to x=1, coming from the positive side, starting at x=1.1 and getting closer to 1, y starts to get very large. At x=1.1, y is 3.23. As x gets closer to 1, say at 1.01, y is 33.22. If we approach 1 from the other side, say from 0.9 to 0.99, y gets very, very large, but negative. What we see is a vertical asymptote at 1, where on one side, y gets very large and positive and on the other side it gets very large, but negative. Let's look at x=-2. As we approach x=-2 from the right-hand side, at -1.9 and -1.99, we see that y gets very large, but negative. If we approach -2 from the left-hand side, like -2.1 and -2.01, we see that y gets very large and positive. Again, we have another vertical asymptote at -2.
Graph for y = 1/(x-1)(x+2); y is undefined at x = 1 and x = -2
In both of our examples, we had vertical asymptotes where our equation was undefined, specifically where we were dividing by zero. This is the key when you're looking for vertical asymptotes.
Let's review. Horizontal asymptotes are where y approaches some number as x goes to positive or negative infinity. This is kind of like our race, where we can only go halfway to the finish line every second. Here, no matter how long we went, as time went to infinity, our distance to the finish would not quite reach zero, but it would get really close. It's approaching zero as time goes to infinity.
A vertical asymptote is where y goes to positive or negative infinity as x approaches some number. This is kind of like as we were approaching that 1-mile marker, we had that hill that just kept getting steeper and steeper and steeper. As we approached the 1-mile marker, y went to infinity.