Limits of sequences and functions - what a limit is, step by step

The idea: as close as you like

“The terms get closer and closer to a number” sounds like a good description of a limit, but it is too vague. The terms of $\frac{1}{n}$ get closer to $-1$ with every step, and yet $-1$ is not their limit: they never come within $1$ of it.

The precise definition works like a challenge:

• someone names a tolerance $\varepsilon$ (the Greek letter epsilon) - any positive number, however small,
• you must answer with a cut-off: from there on, every value is less than $\varepsilon$ away from the limit.

A number is the limit if you can answer the challenge for every $\varepsilon$, however small. For a sequence, the cut-off is a term number $N$: every term from $a_N$ on must be that close. For a function near a point $a$, it is a distance $\delta$ (delta). Every $x$ within $\delta$ of $a$, apart from $a$ itself, must give a value that close.

A band around the limit

Go through the animation step by step with the buttons or the arrow keys. The dots are the terms of $a_n = \frac{n + 1}{n}$: the dot above $n$ is the term $a_n$.

The band is the challenge, and $N$ is the answer to it. A number is the limit when every band, however narrow, has its own $N$, leaving only finitely many terms outside.

Try it yourself: find δ for a function

For a function the challenge is the same, but the answer is a distance from a point $a$. Choose a function and a tolerance $\varepsilon$, which sets the horizontal band around the limit. Then look for a $\delta$, which sets the vertical strip from $a - \delta$ to $a + \delta$. Above the strip, the graph turns green where it stays inside the band and red where it leaves it. A $\delta$ works when no red is left.

What to notice:

• for $\frac{x^2 - 1}{x - 1}$ the largest $\delta$ that works equals $\varepsilon$: away from $x = 1$ the function equals $x + 1$, so its distance from $2$ is exactly $|x - 1|$,
• when you make $\varepsilon$ smaller, $\delta$ has to shrink too - $\delta$ depends on $\varepsilon$,
• $\frac{x^2 - 1}{x - 1}$ and $\frac{\sin x}{x}$ are not defined at the point itself (the graph has a hole there), yet they have limits there, $2$ and $1$ respectively. For $\frac{\sin x}{x}$ with $\varepsilon = 0.1$, any $\delta$ up to about $0.79$ works,
• for $\frac{|x|}{x}$ no $\delta$ works: just left of $0$ the function equals $-1$, and just right of it $1$. A narrow band cannot hold both values, wherever you put it, so this function has no limit at $0$.

The definitions

The limit of a sequence. A number $L$ is the limit of the sequence $(a_n)$ if for every $\varepsilon > 0$ there is a number $N$ such that for every $n \ge N$

We write $\lim\limits_{n \to \infty} a_n = L$, read “the limit of $a_n$ as $n$ tends to infinity is $L$”, and say that the sequence converges to $L$. A sequence without a limit is divergent: $(-1)^n$ keeps jumping between $-1$ and $1$, and $n^2$ grows without bound. For $n^2$ we write $\lim\limits_{n \to \infty} n^2 = \infty$ and say that the sequence diverges to infinity. Infinity is not a number, so this is not a limit in the sense above.

The limit of a function. $\lim\limits_{x \to a} f(x) = L$ if for every $\varepsilon > 0$ there is a $\delta > 0$ such that for every $x$ in the domain of $f$ with $0 < |x - a| < \delta$

The condition $0 < |x - a|$ leaves out $x = a$ itself. That is why the value $f(a)$ does not matter: it can be anything, or $f$ may not be defined at $a$ at all.

Limits through sequences. $\lim\limits_{x \to a} f(x) = L$ exactly when $f(x_n) \to L$ for every sequence $x_n \to a$ with $x_n \ne a$. This equivalent definition, in terms of sequences, is sometimes named after Eduard Heine.

One-sided limits. When $x$ approaches $a$ only from the right, we get the right-hand limit $\lim\limits_{x \to a^+} f(x)$; only from the left, the left-hand limit $\lim\limits_{x \to a^-} f(x)$. The limit exists exactly when both one-sided limits exist and are equal. For $\frac{|x|}{x}$ at $0$ they are $1$ and $-1$.

Limit laws. If $a_n \to A$ and $b_n \to B$, then $a_n + b_n \to A + B$ and $a_n b_n \to AB$, and $\frac{a_n}{b_n} \to \frac{A}{B}$ as long as $B \ne 0$. The same laws hold for limits of functions.

The rest of calculus is built on limits: the derivative, for example, is the limit of the slopes of secants.

Example 1: the limit of a sequence

We want $\lim\limits_{n \to \infty} \frac{2n + 1}{n + 3}$. Both the numerator and the denominator grow without bound, so you cannot read off the answer straight away: this is an indeterminate form of the type $\frac{\infty}{\infty}$.

Example 2: the limit of a function at a hole in its graph

We want $\lim\limits_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Your turn

The function $f(x) = 3x - 1$ has the limit $5$ at $a = 2$. For the tolerance $\varepsilon = 0.3$, find a $\delta$ that works: every $x$ with $0 < |x - 2| < \delta$ must give $|f(x) - 5| < 0.3$.

Common mistakes

• “Closer and closer is enough.” The terms of $\frac{1}{n}$ get closer to $-1$ with every step, but their distance from $-1$ never drops below $1$. The limit is the number they get as close to as you like - here that is $0$.
• “Infinity over infinity is 1.” In $\frac{2n + 1}{n + 3}$ both the numerator and the denominator grow without bound, but the limit is $2$. Divide by the highest power of $n$ first.
• “0/0 means 0.” $\frac{0}{0}$ is an indeterminate form. Factor and cancel first, as in Example 2.
• “The limit is the value at the point.” The limit at $a$ does not look at $f(a)$ at all. $\frac{x^2 - 1}{x - 1}$ is not defined at $1$, yet its limit there is $2$.
• “A sequence never reaches its limit.” It can. The constant sequence $5, 5, 5, \ldots$ has the limit $5$, and every one of its terms equals it.

In exams and at university

In the United States, AP Calculus AB and BC both open with a unit called Limits and Continuity. It uses the informal definition: $f(x)$ can be made as close to the limit as you like by taking $x$ close enough to $a$. The unit covers one-sided limits, the limit laws and simplifying before you substitute, as in Example 2. It also uses the squeeze theorem to show that $\frac{\sin x}{x}$ tends to $1$. The ε-δ definition is not tested on the AP exams, although teachers may cover it if there is time.

In England, A-level Mathematics has no separate topic on limits, but uses them in several places. The gradient of a tangent is a limit, and differentiation from first principles lets $h$ tend to $0$. A geometric series has a sum to infinity only when $|r| < 1$, and integration is understood as the limit of a sum.

The definitions with $\varepsilon$, $N$ and $\delta$ belong mainly to university. Calculus textbooks show them briefly, and a first course in analysis uses them to prove the limit laws.

Check yourself

Flashcards for review

Flip a card and say whether you knew the answer. The ones you did not know come back at the end of the deck. Why it pays to come back to them in a few days is explained in the guide Spaced repetition.