RATIONAL FUNCTIONS Asymptotic Behavior (Page 5 of 5)

Now repeat this process, but this time enter larger and larger positive x values such as x = 5, 10, 100, 1000 and 10000.

 Up arrow key. (5 times)

 5 ENTER key. 10 ENTER key. 100 ENTER key. 1000 ENTER key. 10000 ENTER key.

 Calculator screen image.

From the results we observe that as x approaches positive infinity, the function values get closer and closer to the number 2.5 again, but now are always a little greater than 2.5. We say that y approaches 2.5 from above.

Whenever a function exhibits this type of behavior, where the y values "level off" at some fixed number as x gets very large in the positive or negative sense, we say that the function has a horizontal asymptote at that y value. For this function, there is a horizontal asymptote at y = 2.5.
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