Function study, part 3: limits and asymptotes

Asymptotes are the skeleton of the graph. In the sequence I use to build a function study, they are the first thing I draw on the page, dashed, before I have computed a single derivative: they are the walls the graph has to stay inside, and everything that comes next (sign, increasing and decreasing, concavity) rests on those walls instead of starting from a blank sheet. If you skip this step, or do it badly, you risk drawing a curve that is mathematically consistent with your calculations but visually absurd.

This is the third chapter of the complete guide to function study. In part 2: intercepts and sign study we worked out where the graph lies above or below the x-axis. Here we use limits, but only as a tool, not as a topic in their own right: if you need to revise the actual techniques of computation, indeterminate forms included, it is all in limits explained properly. I take it for granted that you can compute a limit, and I focus on a different question: which limits to compute, and what to do with them once you have them.

Which limits you compute, and why exactly those

You don’t compute limits at random. You compute them at the ends of the domain, and that’s all: nothing else, but nothing less either.

The safest way not to forget any is to write the domain as a union of intervals and read it from left to right. Every “boundary” that appears in that expression generates a limit:

• if the boundary is $-\infty$ or $+\infty$ and the domain really reaches it, you compute the limit as $x\to -\infty$ or $x\to +\infty$;
• if the boundary is a finite number excluded from the domain (a removed point, or the open end of an interval), you must compute the limit from that side. And if the point is a boundary shared between two intervals, you need both sides, right and left, because they are two different limits carrying two different pieces of information.

Let’s take a concrete example. Domain $(-\infty ,1)\cup (1,3)\cup (3,+\infty )$, typical of a function with two excluded points, $x=1$ and $x=3$. The limits to compute are six:

Notice what does NOT appear: no limit as $x\to 2$, because 2 is a perfectly ordinary point of the domain, where the function is defined and continuous. There is nothing to investigate there. The mistake I see most often at this stage is the opposite one: computing limits at interior points of the domain “because the textbook did it in another exercise”, wasting time on information the graph already gives you by itself.

Vertical asymptote

A vertical asymptote is a line $x=x_0$ such that

The natural candidate is every point excluded from the domain for a reason that “blows up”, typically a zero of the denominator. But it is only a candidate: you have to verify it with the limit, not take it for granted.

Why you always need both sides. Take the function

with domain $\mathbb{R}\setminus \{2\}$. Near $x=2$ the denominator changes sign, and the sign of the denominator decides everything:

• as $x\to 2^{+}$, that is $x$ slightly greater than 2 (e.g. 2.01), the denominator $x-2$ is a small positive number, so $\frac{1}{x-2}\to +\infty$;
• as $x\to 2^{-}$, that is $x$ slightly less than 2 (e.g. 1.99), the denominator $x-2$ is small and negative, so $\frac{1}{x-2}\to -\infty$.

Same point, two opposite behaviours. If you compute only one side, you lose half of the information, and when drawing the graph it is precisely that half that tells you whether the curve “plunges” or “soars” on that side.

This is where almost everyone goes wrong: the sign of infinity is not guessed, it is studied. The method is always the same: take the sign of the numerator at the point (if it isn’t zero there, it stays practically constant very close to the point) and the sign of the denominator just to the right and just to the left of $x_0$, then apply the rule of signs. All you need is to mentally substitute a value slightly larger and one slightly smaller than $x_0$ into the denominator and look at the sign that comes out.

One side can be finite. It isn’t certain that both sides blow up: sometimes one of the two limits is a number, and then the vertical asymptote exists only on that side. Take the piecewise function

As $x\to 2^{+}$ we are on the branch $\frac{1}{x-2}$, which we have just seen tends to $+\infty$. As $x\to 2^{-}$ we are on the branch $x$, which simply tends to $2$: a finite limit. Here the vertical asymptote $x=2$ exists only on the right; on the left the graph quietly arrives at the point $(2,2)$ with no blow-up at all.

Not every excluded point is a vertical asymptote. If the limit at an excluded point comes out finite from both sides, you don’t have an asymptote: you have a removable discontinuity, a simple “hole” in the graph. Classic example:

Factorising,

and therefore

A finite limit, the same from both sides: at $x=1$ there is no asymptote, there is a hole in the graph exactly at the point $(1,2)$, where the function “should” pass but is not defined. Always check this before marking a vertical asymptote on the page: a point excluded from the domain guarantees nothing, the limit does.

Horizontal asymptote

If, instead of blowing up, the limit at infinity is a finite number,

The graph gets closer and closer to the horizontal line $y=L$ as $x$ moves away from the origin.

There can be two different ones, one on the right and one on the left: the horizontal asymptote is not a “single” concept for the function, it is a concept per side. A clean example: $f(x)=\arctan x$, defined on all of $\mathbb{R}$. Here

two distinct horizontal asymptotes, $y=\frac{\pi }{2}$ on the right and $y=-\frac{\pi }{2}$ on the left. The graph of $\arctan x$ lives forever inside that horizontal strip, approaching a different “ceiling” on each side.

The graph CAN cross the horizontal asymptote. It is one of the most deeply rooted myths: “an asymptote is a line the function never touches”, repeated as if it were a law of physics. It isn’t: the asymptote describes the behaviour at infinity, it forbids nothing in between. A textbook example:

The limit at infinity is

(numerator bounded between $-1$ and $1$, denominator blowing up, so the ratio goes to zero), horizontal asymptote $y=0$. But $\sin x=0$ at every multiple of $\pi$, so $f(x)=0$ infinitely many times, at $x=\pi ,2\pi ,3\pi ,\dots$. The graph crosses its own asymptote infinitely many times, oscillating ever more tightly around $y=0$ without ever ceasing to touch it. The asymptote is a limiting behaviour, not a ban on contact.

Oblique asymptote

You look for an oblique asymptote only under one precise condition: the limit at infinity is infinite (so no horizontal asymptote on that side) and only on that side. Horizontal and oblique on the same side are mutually exclusive: if there is a horizontal one on the right, you can’t also look for an oblique one on the right, and vice versa. It makes sense: if the limit is finite you already have the answer, nothing more is needed.

When the limit is $\pm \infty$, the procedure is two steps in sequence:

If both limits come out finite, the line $y=mx+q$ is the oblique asymptote. Watch out for the cases where the procedure stops halfway:

• if $m$ doesn’t exist, or comes out $\pm \infty$, there is no oblique asymptote, full stop; there is no point in computing $q$;
• if $m$ is finite but $q$ doesn’t exist or comes out infinite, there is no oblique asymptote all the same (see the example below);
• if $m=0$, you are really falling into the horizontal case ($y=q$), and there is no need to treat it as oblique.

Example of the second case: $f(x)=x+\sqrt{x}$ as $x\to +\infty$. The first step goes fine:

But the second step betrays you:

An infinite $q$ means the function moves further and further away from the line $y=x$ instead of approaching it: no oblique asymptote, even though $m$ was a clean number.

Shortcut for rational functions. If

with $N,D$ polynomials, there is an oblique asymptote exactly when the degree of $N$ exceeds the degree of $D$ by one: not by two, not by three, exactly one. In that case you don’t need to go through $m$ and $q$ with limits: just do the polynomial division, and the quotient is the asymptote.

Complete example:

Degree 2 on top, degree 1 below: the difference is exactly 1, so an oblique asymptote is guaranteed. I divide:

because $x^2+3x+1=(x-2)(x+5)+11$. Therefore

As $x\to \pm \infty$ the term $\frac{11}{x-2}\to 0$, and $x+5$ remains: the oblique asymptote is $y=x+5$, read directly from the quotient of the division, without writing one more limit.

The oblique asymptote too, like the horizontal one, can be crossed by the graph: the same reasoning as before applies. It describes behaviour at infinity, not a local ban.

The case with no asymptotes

Not all functions have them, and that is perfectly fine. A parabola $f(x)=x^2$ goes to infinity with no horizontal asymptote (the limit is infinite) and no oblique one: applying the procedure,

already at the first step $m$ is not finite, so you stop there. The same goes for an exponential like $e^x$ as $x\to +\infty$: it grows too fast to approach any line. In both cases you are not doing anything wrong; it is simply the nature of the function. The mistake would be to insist on looking for an oblique asymptote that doesn’t exist instead of writing “no asymptote as $x\to +\infty$” and moving on.

Branches of a curve with no limit

One last note, without going into too much depth: some functions have no limit at all as $x\to \pm \infty$, neither finite nor infinite. They oscillate forever, like $\sin x$ or $\cos x$. In those cases there is no asymptote on that side, not because the calculation fails but because the behaviour itself “doesn’t converge” towards anything: the function keeps rising and falling without settling. You note it and move on; it is a piece of information in itself, not a hole in the analysis.

The running example of the series: all the asymptotes

Let’s study all the limits and all the asymptotes of

Domain: $\mathbb{R}\setminus \{1\}$, that is $(-\infty ,1)\cup (1,+\infty )$. Ends to check: $-\infty$, $1^{-}$, $1^{+}$, $+\infty$, four limits.

Limits at $x=1$. The numerator at $x=1$ equals $1^2=1$, positive. I look at the sign of the denominator on both sides:

Both infinite, with opposite signs: vertical asymptote $x=1$, with two branches.

Limits at infinity. The degree of the numerator (2) minus the degree of the denominator (1) is exactly 1: I expect an oblique asymptote and no horizontal one. I check with the division: $x^2=(x-1)(x+1)+1$, so

As $x\to +\infty$ and as $x\to -\infty$, $\frac{1}{x-1}\to 0$ in both cases, so

Summary: a single vertical asymptote $x=1$ (with opposite signs on the two branches) and a single oblique asymptote $y=x+1$, valid both for $x\to +\infty$ and for $x\to -\infty$. Four limits computed, two asymptotes found, zero information left on the table: this is exactly the level of completeness I ask of my students on an exam exercise.

The mistakes I see most often

1. Looking for the oblique asymptote when there is already a horizontal one. They are alternatives on the same side, never both. If ${\lim }_{x\to +\infty }f(x)$ is already a finite number, the work on that side ends there.
2. Forgetting a one-sided limit. Every finite excluded point in the domain requires two limits, not one. Skipping one means losing half of the function’s behaviour near that point.
3. Getting the sign of infinity wrong. It is not guessed “by eye”: you study the sign of the denominator just to the right and just to the left of the point, and apply the rule of signs with that of the numerator.
4. Believing the graph can’t touch the horizontal (or oblique) asymptote. False, and $\frac{\sin x}{x}$ proves it: the asymptote is a behaviour at infinity, not a local ban.
5. Forgetting that the two sides can have different asymptotes. A horizontal one on the right doesn’t imply the same on the left (see $\arctan x$), and the same holds for oblique ones: they must be checked separately, always.

How we work on this in lessons

• First we build together the list of limits to compute, starting from the domain written as a union of intervals. It is a mechanical step, but if you skip it you lose asymptotes along the way, so I insist on having it written down every time.
• I insist a great deal on the sign of infinity: I have you redo it out loud until the reasoning “sign of the numerator, sign of the denominator just to the right and just to the left” becomes automatic, not a guess.
• For oblique asymptotes we train both routes, $m$ and $q$ with limits, and polynomial division for rational functions, so you know when the shortcut applies and when it doesn’t.
• With solid foundations on limits and domains, this chapter settles in 2-3 lessons: it is more a matter of method and order than of complicated calculations.

With asymptotes and limits in place, the graph already has its skeleton. The next step is part 4: monotonicity, maxima and minima, where we add the muscle: where the function rises, where it falls, where it stops. If you want to see where you stand on this topic, my whole method is on the maths page, and there is a free introductory call to get started.

The other parts of the series, if you need them out of order: part 1: domain and symmetries, part 5: concavity and inflection points, part 6: the graph and the worked examples. To practise limits in test style, there is the Year 5 test on limits and continuity; the same topic seen from the test’s point of view is in limits, continuity and asymptotes: the Year 5 test. Derivatives, which come next, have a guide of their own: derivatives explained properly.