Historical Context & Motivation
Long before calculators or computers existed, mathematicians wrestled with a deceptively tricky question: what happens to a quantity as it gets closer and closer to some target value, without necessarily reaching it? The ancient Greeks were already bumping into this idea when Zeno of Elea posed his famous paradoxes about motion around 450 BCE. In one paradox, Zeno argued that a runner could never finish a race because they would first have to cover half the distance, then half of the remaining distance, and so on forever. The paradox felt absurd, yet the mathematics to resolve it wouldn't exist for centuries.
The concept of a limit was developed precisely to answer questions like Zeno's. It provides a rigorous way to talk about values that a function or sequence approaches — even if it never actually arrives there. Over the following millennia, this single idea became the bedrock upon which all of calculus is built: derivatives, integrals, and infinite series all rely on limits.
In this lesson, you won't need the full epsilon-delta formalism — that comes later in higher-level courses. Instead, you will learn to investigate limits using two powerful and intuitive tools: graphs and tables of values. Together, these methods will let you determine what value a function is heading toward and decide whether the function is continuous at that point.
Core Principles & Definitions
Before you start calculating limits, you need a clear picture of what the key terms mean. The ideas in this section form the vocabulary you'll use throughout your IB calculus course, so take your time with each one.
Limit of a Function
Left-Hand & Right-Hand Limits
Continuity at a Point
Types of Discontinuity
Visualising Limits on a Graph
The most intuitive way to understand limits is to look at a graph. Below, you'll see a function with several interesting behaviours at different x-values. Pay close attention to the open circle (which means the point is not included) versus the filled dot (which means the point is included). These visual cues tell you whether a function actually takes on a value at that location.
When you read a graph to find a limit, trace the curve from both sides toward the target x-value. If both sides approach the same y-value, that y-value is the limit. Notice in Scenario A that the open circle shows the function doesn't actually equal 2 at x = 1, yet the limit is still 2 because both sides of the curve head toward that height. In Scenario B, the left side and right side approach different y-values, so no single limit exists at x = 3. Scenario C is the ideal case: no holes, no jumps, and the function value matches the limit.
Mathematical Framework
Now let's formalise the ideas from the previous sections using proper mathematical notation. You'll see these expressions throughout your IB exam papers, so becoming comfortable with them is essential.
When you can substitute x = c directly into f(x) and get a real number with no division by zero, no square root of a negative, and no other undefined expression, the limit is simply f(c). This direct-substitution approach works for all polynomial functions and many rational, trigonometric, and exponential functions. When direct substitution gives 0/0 or another indeterminate form, you'll need algebraic techniques like factoring, rationalising, or — later in the course — L'Hôpital's Rule.
Investigating Limits Numerically
When a graph isn't available or when you want more precision, you can estimate a limit using a table of values. The idea is simple: choose x-values that creep closer and closer to c from both sides, compute f(x) for each, and observe the pattern. If the outputs converge toward the same number from both directions, that number is the limit.
Let's investigate the limit of f(x) = (x² − 4)/(x − 2) as x → 2. Substituting x = 2 directly gives 0/0, which is undefined. But the table below reveals what happens as x gets closer to 2.
| x (from the left) | f(x) | x (from the right) | f(x) |
|---|---|---|---|
| 1.5 | 3.5 | 2.5 | 4.5 |
| 1.9 | 3.9 | 2.1 | 4.1 |
| 1.99 | 3.99 | 2.01 | 4.01 |
| 1.999 | 3.999 | 2.001 | 4.001 |
| 1.9999 | 3.9999 | 2.0001 | 4.0001 |
From both sides, the output values converge toward 4. This makes sense algebraically: since x² − 4 factors as (x − 2)(x + 2), the expression simplifies to x + 2 for all x ≠ 2. Substituting x = 2 into x + 2 confirms the limit is 4.
Worked Example
Let's work through a complete problem that combines graphical interpretation, numerical verification, and a continuity check — exactly what you'd see on an IB exam.
Graphical vs. Numerical — Strengths & Limitations
Both graphical and numerical methods are valuable tools for exploring limits, but each has its own strengths and weaknesses. Understanding when to use each method — and when to reach for algebra — will make you much more efficient on the IB exam.
| Criterion | Graphical Method | Numerical Method |
|---|---|---|
| Speed | Very fast — a glance at the graph gives an intuitive answer | Moderate — requires computing several function values |
| Precision | Limited — reading values off a graph is approximate | High — you can get as many decimal places as needed |
| Detecting one-sided limits | Excellent — jumps and holes are visually obvious | Good — requires building separate left/right tables |
| Best for | Identifying types of discontinuity and overall behaviour | Confirming a limit when the graph is unavailable or ambiguous |
| Limitation | Can be misleading if the graph's scale hides subtle behaviour | Cannot prove a limit exists — only provides strong evidence |
Connection to Derivatives & Advanced Theory
Everything you've learned in this lesson about limits is not just a standalone topic — it's the essential foundation for the next major concept in your IB course: the derivative. A derivative is defined as a specific type of limit, so without understanding limits, differentiation would remain a black box.
| This Lesson (SL 5.1) | Coming Next (SL 5.1–5.3) |
|---|---|
| lim f(x) as x → c: what value does f approach? | lim [f(c+h) − f(c)] / h as h → 0: what is the instantaneous rate of change? |
| Evaluate limits using graphs and tables | Evaluate limits using algebraic manipulation (factoring, expanding) |
| Continuity: does the function connect smoothly? | Differentiability: does the function have a well-defined tangent line? |
| One-sided limits detect jumps | One-sided derivatives detect corners and cusps |
Additionally, the concept of continuity carries forward into the Intermediate Value Theorem and later into integration, where you'll calculate areas under curves. The key message is that limits are not a one-time topic — they are the language of calculus, and you'll use them every time you differentiate or integrate. Investing effort now will pay dividends throughout the rest of the IB course.
Practice Problems
Lesson Summary
A limit describes the value that f(x) approaches as x gets arbitrarily close to a target value c. You can investigate limits graphically by tracing the curve from both sides toward c, or numerically by building a table of x-values that creep closer and closer to c. The two-sided limit exists only when both the left-hand limit and the right-hand limit agree on the same value.
A function is continuous at x = c when three conditions are met: f(c) is defined, the limit as x → c exists, and the limit equals f(c). When any condition fails, the function has a discontinuity — which may be a removable hole, a jump, or an infinite discontinuity. These foundational ideas underpin everything in calculus, from derivatives to integrals, making SL 5.1 one of the most important sections in the IB Analysis and Approaches course.