[Calculus Series Part 1] Week 0: Limits of Sequences

This article reopens the calculus series with a refresher guide. Instead of beginning with function limits right away, we step one level earlier and review limits of sequences first. Calculus often looks like a subject about functions, but underneath many of its main ideas sits the question: "Which value do we get arbitrarily close to?" Sequence limits give us the simplest first model for that question.

1. Why start from sequence limits?

Calculus ultimately circles around three questions:

• Which value does some quantity approach?
• How fast does a function change at a moment?
• When a change is broken into tiny pieces, how much accumulates overall?

These map directly to limits, derivatives, and integrals. Sequence limits do not cover every later idea by themselves, but they give us the clearest first setting for understanding what it means to approach a value.

So sequence limits are a useful starting point for limit thinking across the rest of calculus.

2. The very first concept: limits of sequences

2.1 What is the limit of a sequence?

Before using the notation, fix the picture: a sequence is a list of numbers indexed by whole numbers $n=1,2,3,\dots$. For a sequence $a_n$, we ask whether the terms approach some constant $L$ as $n$ becomes large.

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

For example,

$$a_n = \frac{1}{n}$$

gets arbitrarily close to 0 as $n$ increases, so

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

The key point is not that "it eventually equals 0" but that we can make it as close to 0 as we like by taking $n$ large enough.

2.2 Convergent vs. divergent

If a sequence approaches some real number $L$, it converges. If it does not approach any real number, it diverges.

For instance,

$$a_n = n$$

only keeps growing, so it does not converge to any finite real number.

On the other hand,

$$a_n = \frac{2n+1}{n}$$

can be rewritten as

$$\frac{2n+1}{n} = 2 + \frac{1}{n},$$

so

$$\lim_{n \to \infty} \frac{2n+1}{n} = 2.$$

2.3 Oscillation is a different kind of divergence

Divergence is not limited to "blows up to infinity." Some sequences bounce between values without settling down.

For example,

$$a_n = (-1)^n$$

alternates $1, -1, 1, -1, \dots$, so the limit does not exist.

3. From sequences to functions

Once sequence limits feel natural, we can ask a new question: what changes when the input is a real number $x$ rather than an integer index $n$? A sequence moves through a list one term at a time, while a function can be approached through many nearby input values.

The limit of a function tracks where the function value goes as $x$ approaches some number $a$.

$$\lim_{x \to a} f(x) = L$$

For example,

$$f(x) = \frac{x^2 - 1}{x - 1}$$

simplifies to $x+1$ whenever $x \neq 1$, so

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2.$$

But the original expression is undefined at $x=1$. This is a good first example of an important idea: the value of a function at a point and the limit near that point are related, but they are not always the same thing.

3.1 Left-hand and right-hand limits

For a two-sided limit at an interior point, the limit exists exactly when the left-hand and right-hand approaches both exist and agree.

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x).$$

For example,

$$f(x)=\frac{|x|}{x}$$

goes to $-1$ as $x \to 0^-$ and $1$ as $x \to 0^+$, so the limit at 0 does not exist.

3.2 Distinguish divergence to infinity

Sometimes a function value grows without bound instead of approaching a finite number.

$$\lim_{x \to 0^+} \frac{1}{x} = +\infty$$

In such cases we usually say the function tends to $+\infty$ rather than saying it has a finite limit.

4. Concepts right after limits

4.1 Continuity

A function $f(x)$ is continuous at $x=a$ precisely when all three conditions hold:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

Continuity is how we tie the limit back to the actual function value, and it becomes essential when we later check whether a function is differentiable.

4.2 Derivatives

Instantaneous rate of change is defined as the following limit:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}.$$

So a derivative is the limit of average change. Without a solid grip on limits, the definition of the derivative is only symbolic.

4.3 Integrals

Similarly, for suitably well-behaved functions, a definite integral can be defined as the limit of sums whose pieces become finer and finer.

$$\int_a^b f(x)\,dx$$

Before memorizing area formulas, it helps to see the integral as "add up many tiny contributions and take the limit." In many applications this represents accumulated change, not just geometric area.

5. Recommended order for the calculus series

This time we organize the syllabus so the expansion of each concept is visible rather than listing topics flatly. If this section feels long, read it as a map of where the series is going, not as a checklist for today.

I. Building the foundation of limits

Order	Topic	Core idea
1	Limits of sequences	Convergence, divergence, oscillation, basic sequences
2	Laws of sequence limits	Arithmetic rules, inequalities, squeeze patterns
3	Limits of functions	Behavior near a point
4	One-sided limits	Conditions for a full limit to exist
5	Continuity and discontinuity	Connecting limits to actual function values

The goal here is to interpret “getting close” in a mathematically stable way.

II. Moving into differentiation

Order	Topic	Core idea
6	Key function limits	$\sin x / x$, $(e^x-1)/x$, basic exponential and logarithmic limits
7	Difference quotient	From average to instantaneous change
8	Derivatives	Definition and basic computations
9	Basic differentiation rules	Polynomials, exponentials, logarithms, trig
10	Composite functions and the chain rule	Structure of differentiating compositions

Here we want the definition of the derivative to feel like a natural extension of limits.

III. Interpreting and using derivatives

Order	Topic	Core idea
11	Tangents and meaning	Slopes, instantaneous rates, graph insights
12	Increasing vs. decreasing	Reading motion from derivative signs
13	Local maxima and minima	Critical points and extrema
14	Mean value theorem	Linking local change to global change
15	Applications	Sketching graphs, optimization, velocity

Here derivatives become tools for understanding a function, not just computations.

IV. Extending to integrals

Order	Topic	Core idea
16	Indefinite integrals	Viewing integration as the inverse of differentiation
17	Definition of the definite integral	Riemann sums and limits
18	Fundamental theorem of calculus	The bridge between differentiation and integration
19	Basic integration techniques	Substitution and parts
20	Applications	Area, volume, distance, accumulated change

The goal is to treat integration as accumulated change, not just area formulas.

In this structure, the existing notes on exponential/log/trigonometric limits correspond to #6, while trigonometric derivatives and the chain rule align with #9 and #10. We can keep the current articles but add introductory units first so the flow feels natural.

6. Essential practice types for this stage

Treat the items below as a preview of the habits you will need once the series moves from intuition into regular problem-solving.

Type 1. Judge convergence of basic sequences

$$\lim_{n \to \infty} \frac{1}{n}, \qquad \lim_{n \to \infty} \frac{2n+3}{n}$$

Get comfortable deciding where the simplest sequences go.

Type 2. Distinguish divergence vs. oscillation

Do not lump $n$, $(-1)^n$, and $n^2$ into the same "divergent" basket - separate growth from oscillation.

Type 3. Factor and cancel

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$

When substitution gives $0/0$, try factoring before anything else.

Type 4. Compare one-sided limits

Absolute-value and piecewise functions demand that you check the left and right behaviors separately.

Type 5. Use the squeeze theorem

Functions like $x\sin(1/x)$ combine oscillation with damping, so trap them between upper and lower bounds.

Type 6. Apply key standard limits

$$\lim_{x \to 0} \frac{\sin x}{x} = 1, \qquad \lim_{x \to 0} \frac{e^x - 1}{x} = 1$$

These limits are the starting point for derivative formulas, so know what they mean—not just the numbers.

7. Quick checks

Checkpoint 1

Compute $\lim_{n \to \infty} \frac{3n+1}{n}$.

Show answer

Divide first:

$$\frac{3n+1}{n} = 3 + \frac{1}{n},$$

so the limit is 3.

Checkpoint 2

Explain why $a_n = (-1)^n$ has no limit.

Key idea

The terms keep alternating between $1$ and $-1$, so they never settle near a single value.

Checkpoint 3

Find $\lim_{x \to 1} \frac{x^2-1}{x-1}$.

Show answer

Factor the numerator:

$$\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1 \quad (x \neq 1),$$

so

$$\lim_{x \to 1} \frac{x^2-1}{x-1} = 2.$$

8. What’s next?

The next article should flesh out items 1 and 2 from the syllabus: limits of sequences and laws of limits. After that we can move on to function limits, continuity, important standard limits, and finally difference quotients, so the entire calculus arc locks into place.