![]() So, rational expressions may have, but do not necessarily have vertical or horizontal asymptotes. It does have an oblique asymptote of x − 2. But, because the numerator has a higher degree than the denominator, it does not have a horizontal asymptote either. ![]() Some have oblique asymptotes instead.įor example, (x³+5x +1) / (x²+2x + 7) has no vertical asymptote because the denominator has no rational zeros. No, not all rational expressions have vertical or horizontal asymptotes. Note that an oblique asymptote shows how the function behaves, but the function may cross the oblique asymptote before it settles down unlike vertical and horizontal asymptotes which the function approaches but never touches. That is where the oblique asymptote crosses the x-axis. You can find the x-intercept by substituting in a value of zero for the y and solving for x. You can find the equation of that asymptote by dividing your polynomials to find the quotient, which should be of the form y = mx+b - the familiar equation of a straight line with slope m. If the degree of the numerator is greater by two or more, then you do not have a horizontal asymptote -nor an oblique one.įinally! If the degree of the numerator is greater by ONE than the degree of the denominator, then you have an oblique asymptote. ![]() If the degree of the polynomial in the numerator is the same as the degree of the polynomial in the denominator, you will have a horizontal asymptote but no oblique asymptote. An improper rational function has either the same degree or a larger degree in the numerator. ![]() A proper one has the degree of the numerator smaller than the degree of the denominator and it will have a horizontal asymptote. Only improper rational functions will have an oblique asymptote (and not all of those). First you determine whether you have a proper rational function or improper one. The rules for determining that you have an oblique asymptote are complicated and then there are some steps for finding the x-intercept. I am assuming here that you are talking about oblique asymptotes of rational functions and not the oblique asymptotes of hyperbolas (which are not functions). Yes, it will cross the x-axis if you have a slant asymptote (oblique asymptote). ![]()
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