An introduction — limits as approaching behavior, continuity as unbroken function graphs, average and instantaneous rates of change, secant and tangent lines, and area under a curve as a preview of integral calculus
This chapter serves as the bridge between precalculus and Calculus I. Every idea here grows out of something students have already studied: functions, graphs, domain, rate of change, and behavior at the edges of a domain. The goal is not to teach calculus here. The goal is to make calculus feel inevitable — like the natural next question after everything precalculus has built.
Calculus is the mathematics of change and accumulation. Precalculus is the foundation on which calculus stands. Students who understand functions deeply, who can read a graph carefully, who know what a rate of change means, and who have thought about what happens to a function's output as the input approaches a value — those students are genuinely ready for Calculus I.
This chapter introduces three central ideas of calculus as intuition, not formal theory: the limit, continuity, and the rate of change. Students will estimate limits from tables and graphs, interpret one-sided behavior, understand what continuity means and what breaks it, review average rates of change, see how secant lines approach tangent lines, get a first look at instantaneous rate of change as an idea, and preview the concept of area under a curve.
A car is traveling along a highway. Its speed is not constant — it accelerates, decelerates, and changes with traffic. You can measure the car's average speed over the last ten minutes: divide the total distance by ten minutes. But what is the car's speed at this exact instant — right now, at this precise moment?
You cannot measure an instant. An instant has zero length. But you can get closer and closer to an instant by shrinking the time interval: measure speed over five minutes, then two minutes, then thirty seconds, then ten seconds. As the interval shrinks toward zero, the average speed over that interval approaches a number. That number is the instantaneous speed. This is the idea of a limit. This is the beginning of calculus.
Every major idea in calculus — derivatives, integrals, infinite series — depends directly on the function concepts studied in precalculus. Before students can understand what a derivative is, they must understand what a function is, how to evaluate it, how to read its graph, how to compute rates of change, and how to reason about its behavior near special values. Before students can understand integrals, they must be comfortable with area, accumulation, and summation.
Precalculus is not an obstacle before calculus. It is the architecture on which calculus is built.
| Branch of Calculus | Central Question | Key Tool |
|---|---|---|
| Differential Calculus | At any given instant, how fast is this function changing? | The derivative |
| Integral Calculus | How much has accumulated over a period of time or distance? | The integral |
| Common Foundation | As the input approaches a value, what does the output approach? | The limit |
Students have already studied average rates of change in Chapter 1. This chapter will show how that familiar idea leads naturally to instantaneous rate of change — and to calculus. The limit is the bridge that connects the two.
Imagine approaching a building on foot. As you walk toward it, the building gets closer and closer. The concept of a limit is similar: you are asking what value a function's output approaches as the input gets closer and closer to a particular number.
Given a function f(x) and a value x = a, the central question is: as x gets closer and closer to a — from either side — what value does f(x) approach?
The limit of f(x) as x approaches a equals L is written:
lim[x→a] f(x) = L
Read: "The limit of f of x as x approaches a equals L."
This is not asking for the value of f(a). It is asking what value f(x) gets closer and closer to as x approaches a. The input approaches a but may never equal a.
Consider the function f(x) = (x² – 1) / (x – 1). This function is undefined at x = 1 because the denominator equals zero there. But for x ≠ 1, the numerator factors:
f(x) = (x–1)(x+1) / (x–1) = x + 1 for x ≠ 1
So even though f(1) does not exist, the function behaves like y = x + 1 near x = 1. As x approaches 1, the output approaches 1 + 1 = 2. Therefore: lim[x→1] (x²–1)/(x–1) = 2.
The limit of f(x) as x approaches a equals L if the values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a) on either side. A limit is about approaching, not arriving.
One of the most concrete ways to investigate a limit is by constructing a table of values. Students choose inputs that approach the target value from both sides and observe what the outputs approach.
1. Choose input values approaching a from the left (slightly less than a).
2. Choose input values approaching a from the right (slightly greater than a).
3. Compute the output f(x) for each input.
4. If outputs from both sides appear to approach the same value L, then lim[x→a] f(x) = L.
| x | f(x) = (x² – 4) / (x – 2) |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
| x | sin(1/x) |
|---|---|
| 0.1 | sin(10) ≈ −0.544 |
| 0.01 | sin(100) ≈ −0.506 |
| 0.001 | sin(1000) ≈ 0.827 |
| 0.0001 | sin(10000) ≈ −0.306 |
A table can only suggest a limit. It cannot prove one. Use tables as estimation tools, not as definitive proofs. If the table values are not settling clearly toward a single number, suspect that the limit may not exist.
A graph of a function gives visual evidence about limits. Students can trace a graph from either side of a target input value and observe whether the graph appears to approach a single output value from both sides. Crucially, look at where the graph is going — not at any dot that may or may not be marked at x = a.
Draw the graph of y = x + 1 as a straight line with slope 1 and y-intercept 1. At the point (2, 3), place an open circle — indicating that the function is not defined at x = 2. There is no filled dot at x = 2. The graph is a straight line missing exactly one point.
Draw the graph of y = x + 1 for all x ≠ 2, with an open circle at (2, 3). Also draw a filled dot at (2, 5), indicating that g(2) = 5. The isolated filled dot floats above the open circle of the line.
The limit describes where the graph is heading, not where the dot is placed. These two things can be different, and recognizing that difference is the first step in understanding continuity.
Sometimes a function approaches different values from the left and from the right. When this happens, the two-sided limit does not exist, but each one-sided limit may still be defined.
lim[x→a⁻] f(x) — the left-hand limit. As x approaches a from values less than a.
lim[x→a⁺] f(x) — the right-hand limit. As x approaches a from values greater than a.
The two-sided limit lim[x→a] f(x) = L exists if and only if both one-sided limits exist and are equal. If they differ, the two-sided limit does not exist.
Left-hand limit (x → 2⁻): use f(x) = x + 1
lim[x→2⁻] f(x) = 2 + 1 = 3
Right-hand limit (x → 2⁺): use f(x) = 3x − 2
lim[x→2⁺] f(x) = 3(2) − 2 = 4
Draw three pieces: (1) A ray from the left following y = x + 1, approaching the open circle at (2, 3). (2) A filled dot at (2, 5), isolated above the graph. (3) A ray from the right following y = 3x − 2, approaching the open circle at (2, 4). The two open circles sit at different heights, showing clearly that the left and right approaches do not meet.
Some limits describe what happens as the input grows without bound. Others describe what happens as a function's output grows without bound near a particular input. Both types appear in graphs students have already studied.
| Type | Notation | Meaning | Graph Feature |
|---|---|---|---|
| Limit at infinity | lim[x→∞] f(x) = L | As x grows without bound, f(x) approaches L | Horizontal asymptote y = L |
| Limit at −∞ | lim[x→−∞] f(x) = L | As x decreases without bound, f(x) approaches L | Horizontal asymptote y = L |
| Infinite limit | lim[x→a] f(x) = ±∞ | f(x) grows without bound near x = a | Vertical asymptote x = a |
(3x + 1) / (x − 2) = (3 + 1/x) / (1 − 2/x)
As x → ∞: 1/x → 0 and 2/x → 0
lim = (3 + 0) / (1 − 0) = 3
From the right (x → 3⁺):
denominator is small and positive
1 / (small positive) → +∞
From the left (x → 3⁻):
denominator is small and negative
1 / (small negative) → −∞
When we write lim[x→a] f(x) = ∞, we mean the function grows without bound — this is a description of behavior, not a value the limit equals. Technically, such a limit does not exist as a finite number. ∞ is not a number you can substitute or compute with.
A function is continuous if its graph can be drawn without lifting the pencil. That informal description captures something precise: a function is continuous at a point if the function has a value there, if the limit exists there, and if the value and the limit agree.
1. f(a) is defined.
2. lim[x→a] f(x) exists.
3. lim[x→a] f(x) = f(a).
If any one of these three conditions fails, f is not continuous at x = a — it has a discontinuity there.
Condition 1: f(3) = (9 − 9)/(3 − 3) = 0/0 → Undefined. ✗
Since Condition 1 fails, f is not continuous at x = 3.
(Note: the limit does exist — lim[x→3] f(x) = 6 —
but that cannot rescue continuity if f(3) is undefined.)
Condition 1: g(2) = 4 + 6 − 1 = 9 ✓ (defined)
Condition 2: lim[x→2] (x² + 3x − 1) = 9 ✓ (limit exists)
Condition 3: limit (9) = function value (9) ✓
Polynomial, exponential, sine, and cosine functions are continuous everywhere on their domains. For these functions, the limit at any point always equals the function value — no need to verify each condition separately.
When a function fails to be continuous at a point, it has a discontinuity there. There are three main types, each with a distinct graph appearance.
A removable discontinuity occurs when the limit exists, but either f(a) is undefined or f(a) does not equal the limit. This appears as a hole in the graph. The discontinuity is called "removable" because it can be repaired by redefining the function's value at that one point.
Draw the line y = x + 2. At the point (2, 4), place an open circle. The graph is a straight line missing exactly one point. Optionally, a filled dot at (2, 7) would indicate a redefined but still-discontinuous value.
A jump discontinuity occurs when both one-sided limits exist but are not equal. The graph appears to jump from one height to another at x = a.
Draw a horizontal ray at height 2 for x < 1, ending in an open circle at (1, 2). Draw a horizontal ray at height 5 for x ≥ 1, starting with a filled dot at (1, 5). There is a visible vertical gap between the two pieces — the graph literally jumps from height 2 to height 5 at x = 1.
An infinite discontinuity occurs when f(x) grows without bound as x approaches a. The graph has a vertical asymptote at x = a.
Draw the hyperbola y = 1/x. As x → 0⁺, the right branch rises toward +∞. As x → 0⁻, the left branch falls toward −∞. A dashed vertical line marks the y-axis (x = 0) as the asymptote. The function is not defined at x = 0.
| Type | Left Limit | Right Limit | Two-Sided Limit | f(a) Defined? |
|---|---|---|---|---|
| Removable (hole) | L | L | Exists (= L) | No, or f(a) ≠ L |
| Jump | L₁ | L₂ ≠ L₁ | Does not exist | May or may not be |
| Infinite | ±∞ | ±∞ | Does not exist | No |
For each function, determine whether it is continuous at the given point. Identify the type of discontinuity if any condition fails.
f(x) = 3x² − 2x + 5 at x = −1g(x) = (x + 2)/(x − 4) at x = 4h(x) = (x² − 9)/(x − 3) at x = 3f(x) = {x+3 if x<1, 2x+2 if x≥1} at x = 1f(x) = sin(x) at x = πg(x) = 1/(x+2)² at x = −2Students have studied average rate of change throughout precalculus. This section reviews that idea carefully, because it is the direct algebraic ancestor of the derivative.
The average rate of change of f on the interval [a, b] is:
[f(b) − f(a)] / (b − a)
This is the slope of the line connecting the two points (a, f(a)) and (b, f(b)) on the graph of f. It measures how much the output changes per unit of input, averaged across the entire interval.
f(1) = 1, f(4) = 16
Average rate of change = (16 − 1) / (4 − 1) = 15 / 3 = 5
P(1) = 200 · 2¹ = 400
P(3) = 200 · 2³ = 1600
Average rate = (1600 − 400) / (3 − 1) = 1200 / 2 = 600
The line connecting two points on a curve is called a secant line. Its slope equals the average rate of change of the function between those two points.
Given a function f and two distinct input values a and b, the secant line through (a, f(a)) and (b, f(b)) has slope:
m_sec = [f(b) − f(a)] / (b − a)
As the two points are moved closer together — as b is moved toward a — the secant line changes its angle. In the limit, it becomes the tangent line.
f(1) = 1, f(3) = 9
Slope: m = (9 − 1) / (3 − 1) = 8/2 = 4
Point-slope form using (1, 1):
y − 1 = 4(x − 1)
y = 4x − 3
Draw the parabola y = x². Mark the points (1, 1) and (3, 9). Draw the straight line y = 4x − 3 passing through both points. This is the secant line. It crosses the parabola at two locations and passes above the parabola between them.
What happens if the second point moves closer to the first? The table below computes the secant slope for f(x) = x² with a = 1 and decreasing intervals.
| b (second point) | f(b) | Secant Slope = [f(b) − 1] / (b − 1) |
|---|---|---|
| 3 | 9 | 4 |
| 2 | 4 | 3 |
| 1.5 | 2.25 | 2.5 |
| 1.1 | 1.21 | 2.1 |
| 1.01 | 1.0201 | 2.01 |
| 1.001 | 1.002001 | 2.001 |
As b approaches 1, the secant slope approaches 2. This suggests that the slope of the curve at x = 1 — the instantaneous rate of change — is 2. This is the idea of the tangent line.
A tangent line at a point on a curve is the line that just touches the curve at that point and has the same direction (slope) as the curve at that exact location. Unlike a secant line, which crosses the curve at two points, the tangent line touches the curve at one point and represents the curve's local direction there.
As b approaches a, the secant line through (a, f(a)) and (b, f(b)) rotates and approaches a limiting position: the tangent line at x = a. Its slope is:
m_tangent = lim[b→a] [f(b) − f(a)] / (b − a)
In Calculus I, this limit is called the derivative f′(a).
[f(b) − f(1)] / (b − 1)
= (b² − 1) / (b − 1)
= (b − 1)(b + 1) / (b − 1)
= b + 1 for b ≠ 1
lim[b→1] (b + 1) = 2
Draw the parabola y = x². Mark the point (1, 1). Draw the tangent line y = 2x − 1 with slope 2. The tangent line touches the parabola exactly at (1, 1) and runs alongside the curve locally, neither crossing it steeply nor flattening away.
The instantaneous rate of change of a function at a single point describes how fast the function is changing at that exact input — not over an interval, but at a single moment. Limits resolve the apparent paradox of measuring change at a single point.
The instantaneous rate of change of f at x = a is:
lim[h→0] [f(a + h) − f(a)] / h
Here, h is the width of the interval. As h → 0, the two points (a, f(a)) and (a+h, f(a+h)) merge into one, and the secant slope approaches the tangent slope. This limit is the definition of f′(a) in Calculus I.
f(2) = 4 + 1 = 5
f(2 + h) = (2 + h)² + 1 = 4 + 4h + h² + 1 = 5 + 4h + h²
[f(2+h) − f(2)] / h
= [(5 + 4h + h²) − 5] / h
= (4h + h²) / h
= h(4 + h) / h
= 4 + h
lim[h→0] (4 + h) = 4
| Field | Instantaneous Rate of Change Means |
|---|---|
| Physics | Instantaneous velocity — the derivative of position with respect to time |
| Biology | Instantaneous population growth rate |
| Economics | Marginal cost — the derivative of the cost function |
| Medicine | Rate of change of drug concentration in the bloodstream |
| Engineering | Rate of temperature change, rate of current change in a circuit |
Integral calculus studies accumulation: how much of something accumulates as a variable changes. One of the most geometric ways to understand accumulation is through the idea of area under a curve.
Given a function f(x) ≥ 0 on an interval [a, b], what is the total area of the region bounded above by the graph, below by the x-axis, on the left by x = a, and on the right by x = b? For straight-line graphs, this question is straightforward. For curves, it requires limits.
Divide [a, b] into n equal subintervals of width Δx = (b − a)/n. On each subinterval, build a rectangle whose height equals the function's value at a representative input. The area approximation is:
Area ≈ Σ f(xₖ) · Δx (sum from k = 1 to n)
As n → ∞ and Δx → 0, the rectangles fill the region precisely. The exact area is the limit of these Riemann sums, written in Calculus I as:
∫[a to b] f(x) dx
| Subinterval | Right Endpoint | f(x) = x² | Rectangle Area |
|---|---|---|---|
| [0, 0.5] | x = 0.5 | 0.25 | 0.25 × 0.5 = 0.125 |
| [0.5, 1] | x = 1 | 1 | 1 × 0.5 = 0.5 |
| [1, 1.5] | x = 1.5 | 2.25 | 2.25 × 0.5 = 1.125 |
| [1.5, 2] | x = 2 | 4 | 4 × 0.5 = 2.0 |
Total ≈ 0.125 + 0.5 + 1.125 + 2.0 = 3.75
Draw the parabola y = x² from x = 0 to x = 2. Draw four rectangles on [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2], each using the right endpoint height. The rectangles extend above the curve on each subinterval, showing the overestimate. As more rectangles are used, the staircase pattern of rectangle tops approaches the smooth curve.
In Calculus I, students will discover that differentiation and integration are inverse processes. The derivative of the area function (with respect to the upper limit) returns the original function. This relationship is called the Fundamental Theorem of Calculus — one of the most important results in the history of mathematics.
The limit describes approaching behavior, not the value at the point. They may differ. Check all three continuity conditions separately — never assume they agree without verification.
A limit is about approaching, not arriving. The function may approach L without ever equaling L at x = a — for example, when there is a hole at (a, L).
Writing lim[x→a] f(x) = ∞ describes unbounded behavior, not a finite value the limit equals. Technically, the limit does not exist in this case. ∞ is not a number.
Limits apply everywhere. Even perfectly continuous functions have limits at every point — they just happen to equal the function values. Limits are universal; continuity is the special case where limit and value agree.
Average rate of change is the slope of the secant — it describes behavior over an interval. Instantaneous rate is the slope of the tangent — it describes behavior at one exact point. They are related through a limit, but not equal in general.
A Riemann sum is an approximation. The exact area is the limit as the number of rectangles grows without bound. The approximation improves as rectangles become thinner, but it equals the exact area only in the limit.
Estimate each limit using a table or graph description. Show your reasoning.
Determine whether each function is continuous at the given point. State which continuity condition fails, if any, and identify the type of discontinuity.
f(x) = 3x² − 2x + 5 at x = −1g(x) = (x+2)/(x−4) at x = 4h(x) = (x²−9)/(x−3) at x = 3f(x) = {x+3 for x<1, 2x+2 for x≥1} at x = 1f(x) = sin(x) at x = πg(x) = 1/(x+2)² at x = −2f(x) = {x²−1 for x≠2, 5 for x=2} at x = 2f(x) = 3x − 5 on [−2, 4]f(x) = x² − x on [1, 5]f(x) = √x on [4, 9]f(x) = 2^x on [0, 3]f(x) = x² + 1f(x) = x² − 2For Problems 7–8: For f(x) = x², complete the table of secant slopes from x = 3 to x = 3 + h: h = 1, 0.5, 0.1, 0.01, 0.001. Then use the limit definition to find the instantaneous rate of change of f(x) = x² + 4x at x = 1.
Each problem contains a student's work with an error. Identify the error and provide the correct solution.
What is wrong? The form 0/0 is indeterminate — it does not equal 0. What is the correct limit?
What is wrong? Check all three continuity conditions for f(x) = |x| at x = 0. Is this function actually continuous there?
Two errors here: ∞/∞ is indeterminate, and the limit is not 1. What is the correct approach and answer?
What is wrong? An interval containing a discontinuity cannot be called continuous.
What is the error? What would the instantaneous rate of change at x = 2 be? (Use the limit definition.)
Chapter 13 introduced the foundational ideas of calculus — limits, continuity, and rates of change — as intuition built on precalculus. Chapter 14 will close the book by bringing together all function families, modeling techniques, and mathematical reasoning skills developed throughout the course in a capstone review.
In Calculus I, the derivative will be formalized: derivative rules (Power Rule, Product Rule, Chain Rule) will make computation far faster, while the conceptual foundation built here explains why those rules are valid. The Intermediate Value Theorem will use continuity to prove the existence of solutions. The Fundamental Theorem of Calculus will reveal that differentiation and integration — the two great branches of calculus — are inverse operations.
Students who have worked through this textbook arrive at Calculus I knowing functions deeply, reading graphs fluently, working with trigonometric, exponential, logarithmic, and polynomial functions, and understanding the limit as the bridge between algebra and calculus. That readiness is what this book was designed to build.