Let ** f** be a rational function defined as follows

*f*(*x*) = am*x*m + ….. + a1*x* + a0

bn*xm* +….. + b1*x* + b0

**Theorem**

m is the degree of the polynomial in the numerator and n is the degree of the polynomial in the numerator.

**case 1**: For m < n, the horizontal asymptote is the line y = 0.

**case 2**: For m = n, the horizontal asymptote is the line y = am / bn

**case 3**: For m > n, there is no horizontal asymptote

**4.6 The Range**

An examination of the graph above will indicate that all real numbers are available for output. The left most fork of the graph looks like it will rise not quite to y = 1, and the right most fork looks like it may not quite drop to y = 1; so, one might at first suggest that y = 1 is not in the range. However, if you examine the central portion of the graph, you can see that y = 1 is definitely a member of the range. So, all real numbers are present in the range. In interval notation the range would look like this:

(-∞, +∞)

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