Limit of a function

Quantities rarely interest us at a single point; what matters is the approach to it. That is what a limit describes: which number the values $f(x)$ are heading for as the argument closes in on $x_0$ — while the point $x_0$ itself is never taken into account.

Read it as "the limit of $f$ as $x$ tends to $x_0$ equals $g$". Notice that the sentence never says "value" — and that is no accident.

When substitution is enough

For a continuous function — and a polynomial is continuous at every point — the limit simply is the value:

So $\lim_{x \to 3} (x^2 - 2x) = 9 - 6 = 3$, with no trickery involved: the graph has neither a hole nor a jump there, so "heading towards" and "being at" agree. Most limits you will meet work this way, which is why the first move is always to try substituting. The word "continuous" itself is defined later in this lesson, because the definition needs one-sided limits we do not have yet.

The trouble starts only when substitution returns nonsense.

The 0/0 indeterminate form

Look at the function

At $x = 2$ the denominator is zero, so the function has no value there — its domain is every number except two. Substituting gives $\tfrac{0}{0}$, which is nothing at all. And yet the graph looks like this:

Because for every $x \neq 2$ the shared factor may be cancelled:

A limit never looks inside the point $x_0$, so the cancelling is entirely legitimate:

The symbol $\tfrac{0}{0}$ is called indeterminate for exactly this reason: on its own it means nothing. Two other functions landing on $\tfrac{0}{0}$ at a point may have limit $0$, limit $-7$, or none. It is not a result but a note that the computation has to go another way.

Limits at infinity

The second question we put to a function is what happens to it as the argument grows without end. For a rational function the answer is decided by comparing degrees.

Asymptotes — vertical, horizontal and oblique — are taught in rational functions; here we look at one from the side of the limit, because a horizontal asymptote is nothing other than the limit of a function at infinity.

With equal degrees only the ratio of the leading coefficients counts — the remaining terms grow too slowly to change anything:

When the numerator's degree is lower, the denominator grows faster and the quotient dies away:

Formally both cases are one move: divide the numerator and the denominator by the highest power of the denominator and use the fact that $\tfrac{1}{x} \to 0$.

One-sided limits

A point can be approached from two sides — and sometimes that makes a difference. The left-hand limit takes only arguments smaller than $x_0$, the right-hand limit only larger ones:

The ordinary limit exists exactly when both one-sided limits exist and agree:

When there is no limit

The cleanest example is $f(x) = \tfrac{1}{x}$ at zero:

From the right the values grow without bound, from the left they fall without bound:

Limits like these are called improper. Writing $\infty$ does not claim that the limit is some number — it is shorthand for "grows without bound". And since the two sides escape in opposite directions, the limit $\lim_{x \to 0} \tfrac{1}{x}$ does not exist.

Three typical situations in which there is no limit:

• different one-sided limits — the function jumps, the way a tariff changes at midnight;
• escape to opposite infinities — the case of $\tfrac{1}{x}$ above;
• oscillation — the values keep swinging and never settle on a direction.

Continuity at a point

At the start of the lesson a sentence went past: "for a continuous function, substitution is enough". We now have everything needed to say what it means — because continuity is defined through the limit, not the other way round.

A function $f$ is continuous at a point $x_0$ when it meets three conditions at once:

1. it has a value at $x_0$, that is, $x_0$ is in the domain;
2. it has a limit at $x_0$, that is, both one-sided limits exist and agree;
3. that limit equals the value.

In one line:

There are three conditions because each can fail on its own — and every one of those failures has already appeared in this lesson:

• $\tfrac{x^2 - 4}{x - 2}$ at $x_0 = 2$ breaks the first: the limit is $4$, but there is no value at all;
• the courier's price list at $30$ kg breaks the second: there is a value, but the one-sided limits differ;
• a function equal to $x + 2$ everywhere except at $2$, where it stubbornly returns $100$, breaks the third only: limit and value both exist, they simply disagree.

A function is continuous on an interval when it is continuous at each of its points. Polynomials are continuous on the whole of $\mathbb{R}$, rational functions everywhere except the zeros of the denominator, sine and cosine everywhere. And that is the entire content of the sentence we started from: for such functions the limit is substitution, because the third condition says exactly that.

Kinds of discontinuity

Since the conditions can fail in three ways, discontinuities are not all alike either. In each case the name says what can be done about it.

A removable discontinuity — the limit at the point exists, only the value is missing or different. Writing in or correcting a single number makes the function continuous, which is where the name comes from. The graph shows a hole: one missing point, as in $\tfrac{x^2 - 4}{x - 2}$ at $2$.

A jump discontinuity — both one-sided limits exist but differ. No value written in at the point repairs that, because two sides would have to be repaired at once. The graph jumps, the way the courier's tariff does at thirty kilograms.

A discontinuity of the second kind — at least one one-sided limit fails to exist at all: the values escape to infinity, as $\tfrac{1}{x}$ does at zero, or oscillate without settling on a direction.

The first and third cases are exactly the pair that rational functions tells apart as a hole and a pole: when a common factor of numerator and denominator cancels, what is left is a removable discontinuity; when there is nothing to cancel, a vertical asymptote stands there.

The intermediate value property

Continuity is not merely a label that makes arithmetic easier — it carries a theorem no single substitution could reveal.

The intermediate value property (Darboux). If $f$ is continuous on the closed interval $\langle a; b \rangle$, then it takes every value between $f(a)$ and $f(b)$.

The intuition is literally about the picture: the graph of a continuous function is drawn without lifting the pencil, so in travelling from height $f(a)$ to height $f(b)$ no height along the way can be skipped.

What we use most often is the consequence about one intermediate value — zero:

The product is negative exactly when the values at the ends have opposite signs — and then zero lies between them.

Notice what the theorem does not say. It names no $c$, it does not say how many such $c$ there are — there may be many — and it does not hold without continuity: the courier's price list jumps from 22 to 68 and takes no value in between.

What limits are for

They are the only tool that answers the question "how big is the change at a point". The difference quotient — the expression the derivative is about to be built from — heads towards $\tfrac{0}{0}$ as the gap shrinks, so without a limit it cannot be evaluated at all.

The same mechanism lets the region under a curve be sliced into ever thinner strips and pushed to a limit, at which moment the sum of rectangle areas becomes an integral. Both of the lessons that follow rest on this one idea.

Exercises

The exercises cover the three situations of this lesson: the limit of a polynomial (plain substitution), the $\tfrac{0}{0}$ form that needs cancelling, and a limit at infinity. All of them are finite, so the answer is always a number — the cases with $\infty$ and "no limit" are settled by reasoning, not by typing.

Common mistakes

• Treating 0/0 as a result — it is an indeterminate form, an instruction to compute differently, not a number.
• Confusing the limit with the value — a function may have no value at a point and still have a limit there.
• Substituting ∞ into the formula — infinity is not a number; compare degrees or divide by the highest power.
• Declaring a limit to exist after checking one side — both one-sided limits must agree.
• Cancelling without factoring — in $\tfrac{x^2 - 4}{x - 2}$ you may not cancel the bare "x"; factor first, cancel second.
• Writing an improper limit as a number — $+\infty$ describes a behaviour, it is not an answer to type in.
• Confusing the existence of a limit with continuity — a limit may exist where the function has no value; continuity needs both, and needs them equal.
• Applying the intermediate value property to a discontinuous function — the theorem assumes continuity across the interval; without it the graph may jump the axis and have no root at all.