Question

61. Can the graph of a rational function have both a horizontal and an oblique asymptote? Explain.

Short answer

If $m>p$or $m<p$there will be no perfect division occurs, it may have or have not oblique asymptote and it will not have a horizontal asymptote.

Hence, the graph of a rational function can not have both a horizontal and an oblique asymptote.

Step-by-step solution

Step 1 Let us assume a function.

$f\left(x\right)=\frac{a{x}^m+b{x}^{m-1}+.....+k}{a_1{x}^p+b_1{x}^{p-1}+....+k_1}$where $k,k_1$are the constants.

Let $m$be the degree of the numerator.

$p$be the degree of the denominator.

If$m<p$or$m>p$, then there will be no perfect division occurs.

Step 2 If m<p then limx→∞fx becomes 0as a x is left undivided in the denominator.

If $m\ge p+2$then the quotient obtained is a polynomial of degree $2$or higher then this may or may not have a oblique asymptote but it cannot have any horizontal asymptote as no perfect long division occurs.

Hence a single rational function cannot have horizontal as well as oblique asymptote at the same time.