# Describe the end behavior for each of the functions by writing a limit expression

Let's take a look at the function itself f of x. 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.

And when I multiply the bottom I have to distribute that 1 over x times x I get over x.

## End behavior of rational functions

Notice the function approaches negative infinity as x approaches 0 from the left and that it approaches positive infinity as x approaches 0 from the right. In this case we have a positive constant divided by an increasingly small positive number. If you divide out a common factor, then you must also state that the domain does not include the number that would have made the denominator zero. Horizontal Asymptote Definition A horizontal asymptote is a horizontal line that the graph of a function approaches as the magnitude of the input increases without bound in either a positive or negative direction. The limits as or as will be the same if the function has a horizontal asymptote. Some examples are: Note: Divide out all common factors between the numerator and denominator before finding zeros or asymptotes or graphing the function. While there are short-cuts to find the end behavior asymptote in two of these three cases, they all derive from the same procedure. If you divide out a common factor, then you must also state that the domaindoes not include the number that would have made the denominator zero. How might an asymptote be useful?

The graph of the function is a visual representation of the relation decribed by the function rule. This will always be the case when the degree of the denominator is larger than the degree of the numerator. The result will then be an increasingly large positive number and so it looks like the left-hand limit will be positive infinity. Select "Auto" for both Indpnt and Depend. He still trains and competes occasionally, despite his busy schedule. Now this piece goes to 0 so this limit is 10 over or

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