Vertical and Horizontal Asymptotes from the Perspective of Limits

An asymptote is a straight line that the graph of a function approaches ever more closely without, in general, ever touching. Students meet asymptotes informally much earlier, as "lines the curve gets close to", but the previous chapter now supplies the tool to define them with precision: an asymptote is nothing more than a limit statement in geometric disguise. This chapter rebuilds vertical and horizontal asymptotes entirely out of the limit vocabulary already in hand.

To follow it you need the one-sided and infinite limits, the algebra of limits, and the degree-comparison rule for rational functions at infinity, all from the previous chapter. Have a few standard rational and exponential functions in mind — $\frac1x$, $\frac{2x+1}{x-3}$, $e^{-x}$ — as running examples.

1. Vertical Asymptotes Defined Through Infinite Limits

Definition. The line $x = a$ is a vertical asymptote of $f$ if at least one of $\displaystyle\lim_{x\to a^-} f(x)$ or $\displaystyle\lim_{x\to a^+} f(x)$ is $+\infty$ or $-\infty$.

For a rational function $f(x) = \dfrac{p(x)}{q(x)}$ in lowest terms (no common factor left uncancelled), vertical asymptotes occur exactly at the zeros of the denominator, $q(a) = 0$, provided $p(a) \neq 0$. The "in lowest terms" clause matters: if $x=a$ is a zero of both numerator and denominator, it is a candidate for a hole, not an asymptote, until the common factor is cancelled and checked.

Worked example. Locate the vertical asymptotes of $f(x) = \dfrac{x+1}{x^2-x-6}$.

Factor the denominator: $x^2-x-6=(x-3)(x+2)$. Since the numerator $x+1$ shares no factor with $(x-3)(x+2)$, both zeros of the denominator give genuine vertical asymptotes: $x=3$ and $x=-2$. Checking the sign of $f$ near $x=3$: just to the left, $x-3<0$ and $x+2>0$ and $x+1>0$, so $f(x)\to-\infty$; just to the right, all three factors are positive, giving $f(x)\to+\infty$. So

2. Distinguishing a Vertical Asymptote from a Hole

Key distinction. If $x=a$ makes both $p(a)=0$ and $q(a)=0$, factor and cancel first. If the cancelled factor removes the zero entirely, $x=a$ gives a hole, not an asymptote. If, after cancelling, the denominator still vanishes at $a$, a vertical asymptote remains there.

Worked example. Examine $f(x) = \dfrac{x^2-4}{(x-2)^2}$ at $x=2$.

Factor: $f(x) = \dfrac{(x-2)(x+2)}{(x-2)^2} = \dfrac{x+2}{x-2}$ for $x\neq2$. Only one factor of $(x-2)$ cancels; the simplified denominator still vanishes at $x=2$, so $x=2$ remains a vertical asymptote (not a hole), and

is infinite (with opposite signs from each side, since the numerator is positive and the denominator changes sign through $2$).

3. Horizontal Asymptotes Defined Through Limits at Infinity

Definition. The line $y = L$ is a horizontal asymptote of $f$ if $\displaystyle\lim_{x\to\infty} f(x) = L$ or $\displaystyle\lim_{x\to-\infty} f(x) = L$ (or both).

A function can have two different horizontal asymptotes, one as $x\to\infty$ and a different one as $x\to-\infty$; it can have one (the same limit both ways); or it can have none at all, if neither limit exists or both are infinite.

For a rational function, the degree-comparison rule from the previous chapter answers the question completely:

Worked example. Find all horizontal asymptotes of $f(x) = \dfrac{2x-1}{3x+5}$.

Dividing through by $x$ (the highest power in the denominator):

As $x\to\infty$, this tends to $\dfrac23$; as $x\to-\infty$, the same calculation applies, since $\frac1x\to0$ regardless of the sign $x$ approaches infinity from. So $f$ has a single horizontal asymptote, $y = \dfrac23$, on both sides.

4. When the Two Infinite Limits Disagree

Equal-degree rational functions always give the same horizontal asymptote on both sides: if $f(x) = \dfrac{a x^n + \cdots}{b x^n + \cdots}$, then $f(x) \to a/b$ as $x\to\pm\infty$, since dividing through by $x^n$ leaves the ratio $a/b$ regardless of the sign of $x$. Different horizontal behaviour on each side arises instead with surds or exponential functions, where $\sqrt{x^2}=|x|$ forces a sign split.

Worked example. Find the horizontal asymptote(s) of $f(x) = \dfrac{3x}{\sqrt{x^2+1}}$.

For $x\to\infty$ ($x>0$), $\sqrt{x^2+1} = x\sqrt{1+1/x^2}$, so

For $x\to-\infty$ ($x<0$), $\sqrt{x^2+1}=-x\sqrt{1+1/x^2}$ (since $\sqrt{x^2}=|x|=-x$ here), so

So $f$ has two distinct horizontal asymptotes: $y=3$ as $x\to\infty$ and $y=-3$ as $x\to-\infty$.

5. Horizontal Asymptotes of Exponential Functions

The function $f(x) = e^{-x}$ has $\displaystyle\lim_{x\to\infty} e^{-x} = 0$, giving the horizontal asymptote $y=0$ on the right; but $\displaystyle\lim_{x\to-\infty}e^{-x} = \infty$, so there is no horizontal asymptote on the left. More generally, $f(x) = ka^{-x}+c$ (with $0<a<1$ excluded, $a>1$ assumed, $k\neq0$) has a single horizontal asymptote $y=c$ as $x\to\infty$ only, reflecting the one-directional decay of exponential functions — a fact used repeatedly in later applications such as cooling curves and population models.

6. A Function Can Cross Its Horizontal Asymptote

Common misconception corrected. A horizontal asymptote describes the limiting behaviour of $f$ as $x\to\pm\infty$ — nothing about the value of $f$ at finite $x$. A graph may cross its own horizontal asymptote any number of times before eventually levelling off towards it.

For instance, $f(x) = \dfrac{\sin x}{x} + 1$ has horizontal asymptote $y=1$ (since $\frac{\sin x}{x}\to0$), yet it oscillates across the line $y=1$ infinitely often as $x\to\infty$, each oscillation shrinking because $\left|\frac{\sin x}{x}\right|\leq\frac1x\to0$. A vertical asymptote, by contrast, is never crossed, since the function is not even defined at $x=a$.

7. Worked Synthesis: Full Asymptote Analysis

Worked example. Fully describe the asymptotes of $f(x) = \dfrac{2x^2-8}{x^2-x-2}$.

Factor both: numerator $2x^2-8 = 2(x-2)(x+2)$; denominator $x^2-x-2=(x-2)(x+1)$. A common factor of $(x-2)$ cancels:

• At $x=2$: the factor cancelled entirely, so there is a hole, not an asymptote. Its height is $\frac{2(4)}{3}=\frac83$, giving the point $\left(2,\frac83\right)$.
• At $x=-1$: the simplified denominator still vanishes, and the numerator $2(x+2)=2\neq0$ there, so $x=-1$ is a genuine vertical asymptote.
• As $x\to\pm\infty$: numerator and simplified denominator both degree $1$, so the limit is the ratio of leading coefficients, $\dfrac21=2$ on both sides — a single horizontal asymptote, $y=2$.

Common Exam Pitfalls

• Finding zeros of the denominator before checking for common factors. A zero shared with the numerator is a hole candidate, not automatically a vertical asymptote — always factor and cancel first.
• Assuming equal-degree rational functions always give identical asymptotes on both sides. They do — dividing every term by $x^n$ leaves the ratio of leading coefficients whatever the sign of $x$ — but the habit of assuming it is dangerous: surd or exponential expressions routinely give different left and right horizontal behaviour.
• Believing a graph can never cross a horizontal asymptote. It can, and often does, provided it still settles towards the line as $x\to\pm\infty$; only vertical asymptotes are never crossed (the function is undefined there).
• Forgetting that a function may have two horizontal asymptotes, one, or none. Each direction ($x\to\infty$ and $x\to-\infty$) must be checked as a separate limit calculation.
• Mishandling $\sqrt{x^2}$ as $x\to-\infty$. Writing $\sqrt{x^2}=x$ unconditionally is the single most common sign error in finding asymptotes of functions containing surds.
• Citing the vertical asymptote without stating the one-sided signs. A complete answer states whether $f\to+\infty$ or $f\to-\infty$ from each side, not merely that "there is an asymptote at $x=a$".

Summary

• A vertical asymptote at $x=a$ occurs when $\displaystyle\lim_{x\to a^-}f(x)$ or $\displaystyle\lim_{x\to a^+}f(x)$ is $\pm\infty$; for a rational function in lowest terms, this happens exactly at the zeros of the denominator.
• A zero shared by numerator and denominator gives a hole once cancelled, unless the denominator still vanishes afterwards, in which case the vertical asymptote survives.
• A horizontal asymptote $y=L$ occurs when $\displaystyle\lim_{x\to\infty}f(x)=L$ or $\displaystyle\lim_{x\to-\infty}f(x)=L$; a function may have two, one, or none.
• For rational functions, the degree-comparison rule gives the horizontal asymptote directly: $0$ if $\deg p <\deg q$, the leading-coefficient ratio if equal, none if $\deg p > \deg q$.
• Surds of the form $\sqrt{x^2+\cdots}$ and exponential functions typically give different horizontal asymptotes as $x\to\infty$ and $x\to-\infty$, because of the $\sqrt{x^2}=|x|$ sign split or the one-directional decay of $e^{-x}$.
• A graph can cross its horizontal asymptote arbitrarily often; it can never cross a vertical asymptote, since the function is undefined there.