4.5.2.1 How to find the horizontal asymptote?

Let f be a rational function defined as follows

f(x) = amx­m + ….. + a1x + a0

bnxm +….. + b1x + 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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