Limit of a Sequence: Definition, Rules & Examples

What Is The Limit Of A Sequence?

The limit of a sequence is the value its terms approach as the index $n$ runs to infinity. A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$, one for each positive integer, and the number $a_n$ is called the $n$-th term. When those terms crowd arbitrarily close to a single finite value $L$, we say the sequence converges and write:

$$\lim_{n \to \infty} a_n = L$$

Read this as "the limit of $a_n$, as $n$ tends to infinity, equals $L$." The idea is not that any term equals $L$, but that you can force the terms as close to $L$ as you please by going far enough out in the list.

Take $a_n = \dfrac{1}{n}$. The terms run $1, \tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \dots$, shrinking toward $0$ without ever landing on it. The limit is $0$, so this sequence is convergent. Compare it with the neighbouring idea of a limit of a function: there the input slides continuously toward a point, while here the input jumps through the whole numbers only.

What Is The Formal Definition Of The Limit Of A Sequence?

The intuitive picture ("terms get close to $L$") needs a precise test, because "close" and "eventually" have to mean something exact. That test is the $\varepsilon$–$N$ definition.

$$\lim_{n \to \infty} a_n = L \quad \text{means} \quad \text{for every } \varepsilon > 0 \text{ there is an } N \text{ such that } |a_n - L| < \varepsilon \text{ for all } n \ge N.$$

Unpack it in plain language. Someone challenges you with a tolerance $\varepsilon$, a narrow band of half-width $\varepsilon$ drawn around $L$. To win, you must name an index $N$ so that every term from $a_N$ onward lands inside that band. If you can always answer, no matter how tiny the $\varepsilon$, then $L$ is the limit.

For $a_n = \dfrac{1}{n} \to 0$, the challenge is easy to meet. Given $\varepsilon > 0$, choose any $N$ larger than $\dfrac{1}{\varepsilon}$. Then for $n \ge N$,

$$\left| \frac{1}{n} - 0 \right| = \frac{1}{n} \le \frac{1}{N} < \varepsilon.$$

Every term past that index sits inside the band, so the definition is satisfied and the limit is confirmed to be $0$.

What Is The Difference Between Convergent And Divergent Sequences?

A sequence is convergent if it has a finite limit $L$, and divergent if it does not. Divergence covers three different behaviours, and mixing them up is a common source of error.

• Growing without bound. The terms head to infinity, as with $a_n = 2n$ or $a_n = n^2$. These have no finite limit, so they diverge. Some texts write $a_n \to \infty$, but this is shorthand for a specific kind of divergence, not a finite limit.

• Oscillating. The terms bounce between values and never settle, as with $a_n = (-1)^n$, which alternates $-1, 1, -1, 1, \dots$. It stays bounded yet still diverges, because it approaches no single value.

• Wandering. The terms neither settle nor march to infinity in a clean way, so no limit exists.

The middle case carries the trap: a sequence can be perfectly bounded and still divergent. Staying finite is not the same as converging. To converge, the terms must close in on one value, which $(-1)^n$ never does.

What Are The Limit Laws For Sequences?

Once you know a few limits, the limit laws let you build others without returning to the $\varepsilon$–$N$ definition each time. Suppose $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, both finite. Then the combined sequences converge as you would hope, and the full set is collected in the limit laws reference.

• Sum and difference: $\lim_{n \to \infty} (a_n \pm b_n) = A \pm B$.

• Scalar multiple: $\lim_{n \to \infty} (c,a_n) = cA$ for any constant $c$.

• Product: $\lim_{n \to \infty} (a_n,b_n) = AB$.

• Quotient: $\lim_{n \to \infty} \dfrac{a_n}{b_n} = \dfrac{A}{B}$, provided $B \neq 0$.

Every law carries the same fine print: both pieces must converge first. The laws describe how finite limits combine, and applying them to a piece that diverges is exactly where marks are lost, a slip the mistakes section returns to.

Example 1: A ratio of polynomials in $n$.

Find $\lim_{n \to \infty} \dfrac{3n + 1}{n + 2}$.

Divide the top and bottom by the highest power of $n$, here $n$ itself:

$$\frac{3n + 1}{n + 2} = \frac{3 + \frac{1}{n}}{1 + \frac{2}{n}}.$$

As $n \to \infty$, both $\dfrac{1}{n} \to 0$ and $\dfrac{2}{n} \to 0$, so by the quotient and sum laws the expression tends to $\dfrac{3 + 0}{1 + 0} = 3$.

Final answer: $\lim_{n \to \infty} \dfrac{3n + 1}{n + 2} = 3$.

Example 2: A divergent sequence.

Test $a_n = (-1)^n$ for convergence.

The terms are $-1, 1, -1, 1, \dots$. Suppose a limit $L$ existed; then every term past some index would have to sit within $\varepsilon = \tfrac{1}{2}$ of $L$. But consecutive terms are always a distance $2$ apart, so they cannot both lie in a band of width $1$, and no such $L$ exists.

Final answer: $(-1)^n$ diverges; it has no limit.

How Does The Squeeze Theorem Work For Sequences?

The squeeze theorem handles sequences that are too tangled to attack directly, by trapping them between two simpler sequences that share a limit. It is the sequence version of the same idea used for a squeeze theorem on functions.

If $a_n \le b_n \le c_n$ for all $n$ past some index, and

$$\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L,$$

then the trapped sequence is forced to the same value: $\lim_{n \to \infty} b_n = L$.

Example 3: A sequence with an oscillating numerator.

Find $\lim_{n \to \infty} \dfrac{\sin n}{n}$.

The sine term wanders between $-1$ and $1$, so the sequence itself has no clean formula for its behaviour. Bound it instead. Since $-1 \le \sin n \le 1$, dividing by the positive number $n$ gives:

$$-\frac{1}{n} \le \frac{\sin n}{n} \le \frac{1}{n}.$$

Both outer sequences, $-\dfrac{1}{n}$ and $\dfrac{1}{n}$, converge to $0$. The squeeze theorem forces the middle sequence to $0$ as well.

Final answer: $\lim_{n \to \infty} \dfrac{\sin n}{n} = 0$.

How Is The Limit Of A Sequence Related To The Limit Of A Function?

Sequences and functions are tied together by a bridge theorem that lets you borrow the whole machinery of function limits, including L'Hôpital's rule.

The bridge theorem. If $a_n = f(n)$ for some function $f$, and $\lim_{x \to \infty} f(x) = L$, then $\lim_{n \to \infty} a_n = L$.

In words: if the continuous function settles to $L$ as $x$ runs to infinity, the sequence sampled at the whole numbers settles to the same $L$. This connects directly to limits at infinity and the collection of standard limits you already use for functions.

The bridge runs one way only, because a sequence can converge even when the matching function has no limit. Take $a_n = \sin(2\pi n)$. Every term is $\sin(2\pi n) = 0$, so the sequence converges to $0$, yet $f(x) = \sin(2\pi x)$ oscillates forever and has no limit at infinity. You may pass from the function to the sequence, never the reverse.

Example 4: Using the function bridge with L'Hôpital.

Find $\lim_{n \to \infty} n^{1/n}$.

The form is an indeterminate $\infty^0$, so take the natural log and treat $n$ as a continuous variable $x$. Let $f(x) = \dfrac{\ln x}{x}$. Applying L'Hôpital's rule to the $\tfrac{\infty}{\infty}$ form, differentiate top and bottom: $f'$ compares $\dfrac{1/x}{1}$, so

$$\lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{1}{x} = 0.$$

The log of the sequence tends to $0$, so the sequence itself tends to $e^0 = 1$.

Final answer: $\lim_{n \to \infty} n^{1/n} = 1$.

What Are The Most Common Limits Of Sequences?

A handful of sequences appear so often that their limits are worth memorising. Each entry below is verified, and the whole set is used constantly once series enter the picture.

Table: Standard limits of sequences, with the condition that makes each valid.

Two rows deserve a note. For $r^n$ with $\lvert r \rvert < 1$, even a negative ratio such as $r = -\tfrac{1}{2}$ still gives $r^n \to 0$, because the size $\lvert r \rvert^n$ shrinks and drags the whole term to zero regardless of the alternating sign. And $\left(1 + \tfrac{1}{n}\right)^n$ does not run off to infinity despite the growing exponent: it climbs steadily and is bounded above, settling on $e \approx 2.7183$.

Why Does The Limit Of A Sequence Behave This Way?

The definitions can feel abstract until you see what forces each result. The behaviour comes from a race between the parts of the term.

• A shrinking piece beats a constant. In $\dfrac{1}{n}$ or $\dfrac{c}{n}$, the denominator grows without limit while the top stays fixed, so the quotient is crushed to $0$. This single fact drives most first limits.

• Repeated multiplication by a small factor collapses. For $\lvert r \rvert < 1$, each step multiplies by something smaller than $1$ in size, so $\lvert r \rvert^n$ falls geometrically. By the same reasoning, $\lvert r \rvert > 1$ multiplies up each time and the terms escape to infinity.

• Boundedness plus one-way motion pins a limit down. The sequence $\left(1 + \tfrac{1}{n}\right)^n$ increases at every step yet never passes $3$. A sequence that only climbs and is capped from above has nowhere to go but toward a ceiling value, which is the monotone-convergence idea behind monotonic sequences.

Seen this way, a limit is the outcome of which part of the term wins the race as $n$ grows. Convergence happens when the changing pieces are tamed, either by shrinking to nothing or by being boxed in from one side.

Who Shaped The Idea Behind The Limit Of A Sequence?

Sequences that creep toward a value were computed for centuries before anyone wrote down what "creep toward" precisely meant.

Two later figures supplied the rigour the idea had been missing:

• Augustin-Louis Cauchy (1789–1857, France) built limits into the foundation of analysis in his 1821 Cours d'analyse, defining convergence in terms of terms becoming and staying arbitrarily close.

• Karl Weierstrass (1815–1897, Germany) sharpened Cauchy's language into the exact $\varepsilon$–$N$ form used today, removing the last appeals to intuition and motion.

Where Is The Limit Of A Sequence Used In The Real World?

The idea reaches well past the classroom, because so many processes are really a sequence of steps homing in on a final value.

• Finance: continuous compounding is the limit of $\left(1 + \tfrac{r}{n}\right)^n$ as compounding grows more frequent, which is why $e$ sits at the heart of interest and growth formulas.

• Numerical methods: Newton's method for solving an equation produces a sequence of approximations $x_1, x_2, x_3, \dots$ whose limit is the root, and each new term is closer than the last.

• Computer science: fixed-point iteration, from page-ranking to solving physics on a grid, runs a rule repeatedly until the sequence of outputs stops changing, meaning it has reached its limit.

• Physics and engineering: a decaying quantity such as charge on a capacitor or a cooling temperature is sampled as a sequence that converges to a steady state, a direct $r^n$-style decay.

• Probability: long-run averages and simulation estimates converge to a true value as the number of trials grows, the practical face of a limit.

One idea, the value a list of numbers approaches, underwrites banking, root-finding algorithms, and physical steady states alike. Wherever a process repeats and settles, a limit of a sequence is the number it settles on.

What Are The Most Common Mistakes With The Limit Of A Sequence?

These four errors account for most lost marks, and each matches a question real students ask on r/calculus, r/learnmath, and in the research literature on how learners picture limits.

Assuming every bounded sequence converges.

Where it slips in:

A student sees that $(-1)^n$ never runs off to infinity and concludes it must have a limit because it stays between $-1$ and $1$.

Don't do this:

Do not treat "stays finite" as "converges." Divergence is not only about racing to infinity.

The correct way:

Check that the terms close in on one value. Because $(-1)^n$ keeps jumping a distance $2$ between consecutive terms, no single $L$ can trap them, so it diverges despite being bounded.

Guessing the limit of $n^{1/n}$ from the pieces.

Where it slips in:

A student reasons that the base $n$ runs to infinity, so the sequence must diverge, or that the exponent $\tfrac{1}{n}$ runs to $0$, so the answer must be $1$ only by luck.

Don't do this:

Do not read off an indeterminate $\infty^0$ form by eyeballing the base and the exponent separately.

The correct way:

Take the log and resolve it honestly. Since $\dfrac{\ln n}{n} \to 0$, the sequence tends to $e^0 = 1$, a result you must derive rather than guess.

Splitting a limit law across a divergent term.

Where it slips in:

A student writes $\lim (a_n - b_n) = \lim a_n - \lim b_n$ when both $a_n$ and $b_n$ tend to infinity, and reports $\infty - \infty = 0$.

Don't do this:

Do not apply the sum, product, or quotient law unless each piece has its own finite limit.

The correct way:

Rewrite the term first so the pieces converge. For example, $\sqrt{n+1} - \sqrt{n}$ is an $\infty - \infty$ form; multiplying by the conjugate turns it into $\dfrac{1}{\sqrt{n+1} + \sqrt{n}}$, which clearly tends to $0$.

Forgetting the $\lvert r \rvert < 1$ condition on $r^n$.

Where it slips in:

A student sees the alternating signs of $\left(-\tfrac{1}{2}\right)^n$ and calls it divergent, or applies $r^n \to 0$ to $r = 2$ and reports $0$.

Don't do this:

Do not judge $r^n$ by its sign or by habit. The size $\lvert r \rvert$ decides everything.

The correct way:

Compare $\lvert r \rvert$ to $1$. Here $\left| -\tfrac{1}{2} \right| = \tfrac{1}{2} < 1$, so $\left(-\tfrac{1}{2}\right)^n \to 0$, while $2^n$ has $\lvert r \rvert = 2 > 1$ and diverges.

Practice Problems On The Limit Of A Sequence

Work each one, then check against the answer. Answers are verified.

1. Find $\lim_{n \to \infty} \dfrac{5n - 2}{3n + 4}$.
   (Answer: divide by $n$; $\dfrac{5 - 2/n}{3 + 4/n} \to \dfrac{5}{3}$.)

2. Find $\lim_{n \to \infty} \dfrac{7}{n^2}$.
   (Answer: the denominator grows without bound, so the limit is $0$.)

3. Find $\lim_{n \to \infty} (0.9)^n$.
   (Answer: $\lvert 0.9 \rvert < 1$, so the limit is $0$.)

4. Find $\lim_{n \to \infty} \dfrac{2n^2 + 1}{n^3}$.
   (Answer: split as $\dfrac{2}{n} + \dfrac{1}{n^3} \to 0$.)

5. Find $\lim_{n \to \infty} \dfrac{(-1)^n}{n}$.
   (Answer: squeeze between $-\tfrac{1}{n}$ and $\tfrac{1}{n}$; the limit is $0$.)

6. Does $\lim_{n \to \infty} \left(1 + \dfrac{3}{n}\right)^n$ exist, and what is it?
   (Answer: yes; it is $e^3 \approx 20.0855$.)

Where Should You Go Next After The Limit Of A Sequence?

The limit of a sequence is the gateway to the whole of infinite processes, and several natural doors open from here.

1. Monotonic sequences. Learn the monotone-convergence theorem, the exact reason a bounded, always-climbing sequence such as $\left(1 + \tfrac{1}{n}\right)^n$ must have a limit.

2. Limit of a function. Strengthen the continuous version of the idea and the bridge theorem that lets you use L'Hôpital's rule on sequences.

3. Convergence and divergence of series. A series converges exactly when its sequence of partial sums has a limit, so this is the direct next step into infinite series.

If your child is meeting the limit of a sequence for the first time, a live Bhanzu trainer teaches it from the approaching-a-value picture up, so convergence feels concrete, in the Bhanzu math program.