When we evaluate limits of a function as (x) goes to infinity or minus infinity, we are examining something called the end behavior of a limit. In order to determine the end behavior, we need to substitute a series of values or simply the function determine what number the function approaches as the range of the function increases or decreases towards infinity or minus infinity.
I want to talk about limits and End Behavior for functions. Let's take a look at a function f of x equals 10x over x-2. What happens as x goes to infinity, now I can't plug infinity into this function to find out but I can take limits as x approaches infinity. Let's take a look at the function itself f of x. Now right now 10x over x-2 you can see that both the numerator and denominator are going to get very very big as x moves towards infinity. So that's it's not clear exactly what's happening but one thing that can make it clear is a little Algebraic trick that gets rid of the x's, I'm going to multiply the top and bottom by 1 over x. This is usually not a desired move but here it's going to be very useful when I multiply the top by 1 over x the x's cancel and I'm left with 10. And when I multiply the bottom I have to distribute that 1 over x times x I get 1-2 over x. It's a lot easier to see what happens as x goes to infinity when you look at the function written in this form right, this is exactly the same function as this. But in this form you can see that as x goes to infinity this term is going to 0 and so the rest is going to 10 over 1-0. So the limit as x approaches infinity of f of x is the limit as x approaches infinity of 10 over 1-2 over x. Now this piece goes to 0 so this limit is 10 over 1-0 or 10. Let's try this trick and the limit as x approaches negative infinity. Now we can still use the fact that f of x equals 10 over 1-2 over x, so the limit as x approaches negative infinity of f of x is the limit as x approaches negative infinity of 10 over 1-2 over x that's a negative. Now as x approaches negative infinity this quantity is still going to go to 0, so this limit is still 10 over 1-0 or 10 oops 10. Now what does it mean that it approaches that the function's value approaches 10 as x approached infinity or as x approaches minus infinity. Well let's take a look at the graph. As x approaches infinity the value of the function approaches 10 and that means it approaches this horizontal line y=10. That indicates the presence of a horizontal asymptote and the same thing as x approaches negative infinity the values are approaching 10 they're approaching from below but they're still headed towards this number 10. And so both of those limits indicate that y=10 is a horizontal asymptote and here is our definition of horizontal asymptote, y=b is a horizontal asymptote for the graph of a function y=f of x if the limit is x approaches infinity of f of x is b or if the limit is x approaches negative infinity of f of x=b you only need one of these to be true in order for y=b to be a horizontal asymptote.