How do you find the slant asymptotes of rational functions?


Quick Answer

In order to find the slant asymptote of a rational function, the student must divide the numerator by the denominator. The most common division methods used are long division and synthetic division, and most teachers instruct their students to use long division. A crucial step to finding the asymptote is to rearrange the rational function according to the results obtained by long division so that the polynomial part is separated.

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Full Answer

In most cases, the results of the long division produce a polynomial that has a similar graph to the original function. These graphs indicate that there is only a slight difference between the graph of a function and its slant asymptote if the difference in their degrees is 1.

An asymptote is a line similar to the x-axis or the y-axis that the curve approaches but does not cross. Other types of asymptotes that relate to rational functions are the horizontal and vertical asymptotes. A vertical asymptote is calculated by finding the roots of q(x) without involving the numerator, as only the denominator is used in the calculation. Horizontal asymptotes are located only to the far right or the far left of the graph, and they are not asymptotic at the center.

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