If a rational function has no horizontal asymptote, does it then have to have a slant asymptote

Question

This is assuming that the function is in a fractional form where the the degree of the numerator is higher than the degree of the denominator.

Answer 1

Let $f(x)=P(x)/Q(x)$ be a rational function, where $P$ and $Q$ are nonzero polynomials; then

• $f$ has a horizontal asymptote if and only if $\deg P\le\deg Q$

• $f$ has an oblique asymptote if and only if $\deg P=1+\deg Q$

Proof. We can write $P(x)=A(x)Q(x)+R(x)$, where $\deg R<\deg Q$ (or $R(x)$ is the zero polynomial). Thus $$ f(x)=A(x)+\frac{R(x)}{Q(x)} $$ Clearly $$ \lim_{x\to\pm\infty}\frac{R(x)}{Q(x)}=0 $$ Thus, if $\deg A(x)>0$ (that is, $\deg P>\deg Q$), $\lim_{x\to\pm\infty}f(x)=\pm\infty$ (with suitable choices of sign in the limit). If $A(x)=c$ is constant, then $$ \lim_{x\to\pm\infty}f(x)=c $$ and so $f$ has a horizontal asymptote. This is the same as saying that $\deg P\le\deg Q$.

Suppose $\deg P=1+\deg Q$; then $A(x)=ax+b$, with $a\ne0$. Then $$ \lim_{x\to\pm\infty}\frac{f(x)}{x}=a $$ and $$ \lim_{x\to\pm\infty}(f(x)-ax)=b $$ If $\deg P>1+\deg Q$, then $$ \lim_{x\to\pm\infty}\frac{f(x)}{x}=\pm\infty $$ (again with suitable choices of signs in the limit) and $f$ has no oblique asymptote.

Answer 2

No, a slant asymptote occurs when the numerator has a degree exactly one unit higher than the denominator. If the difference between degrees is greater, then there cnb be only vertical asymptotes.