Historical Context & Motivation
Long before anyone invented modern calculus, mathematicians wrestled with a fundamental question: what happens to a quantity as it gets closer and closer to some target value, without necessarily reaching it? Ancient Greek thinkers encountered this puzzle when they tried to compute the area of a circle by filling it with polygons that had more and more sides. The idea of approaching a value — getting infinitely close — is the seed from which the concept of a limit grew.
The central question this topic addresses is deceptively simple: if you cannot plug a value directly into a function (perhaps because it causes division by zero), can you still figure out what the function wants to equal? Limits give us a reliable, systematic way to answer that question using graphs and tables of values — the two approaches you will master in this lesson.
Core Principles & Definitions
Before diving into calculations, let's pin down the key ideas. A limit is not about what a function equals at a point; it is about what the function approaches as the input gets closer and closer to that point. This distinction matters because sometimes a function has no value at a point (like a hole in the graph), yet the limit can still exist.
Limit of a Function
One-Sided Limits
Continuity
Discontinuity
Graphical Approach to Limits
One of the most powerful ways to understand limits is to look at them on a graph. The diagram below shows three common situations you will encounter: a function that is continuous at a point, a function with a hole (removable discontinuity), and a function with a jump discontinuity. By tracing the curve from the left and from the right, you can visually identify whether a limit exists.
When you read a limit from a graph, follow this process. First, place your finger on the curve to the left of the target x-value and slide it towards the point — note the y-value you are approaching. Then do the same from the right. If both sides converge to the same y-value, the limit exists and equals that value. If they converge to different y-values, the two-sided limit does not exist.
Mathematical Framework — Notation & Numerical Approach
In this section, we formalise the notation and explore the numerical (table) method for estimating limits. The notation may look intimidating at first, but it simply condenses the graphical ideas from the previous section into compact symbols.
The Numerical (Table) Method
When you cannot evaluate f(a) directly — for example when substituting a leads to 0/0 — you can build a table of values with inputs that creep closer and closer to a. Choose values from both sides: for x → 2, you might try 1.9, 1.99, 1.999 from the left and 2.1, 2.01, 2.001 from the right. If the outputs settle towards a single number, that number is your limit estimate.
| x (approaching 2⁻) | f(x) = (x² − 4)/(x − 2) | x (approaching 2⁺) | f(x) = (x² − 4)/(x − 2) |
|---|---|---|---|
| 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 |
Even though f(2) is undefined (we get 0/0), the table clearly shows the outputs converging to 4 from both sides. Therefore, lim(x→2) (x² − 4)/(x − 2) = 4. You can verify this algebraically by factoring: (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2, and substituting x = 2 gives 4.
Classifying Discontinuities
Understanding the different ways continuity can fail helps you identify limits quickly. There are three main types of discontinuity you will meet in IB Applications and Interpretation, and each one has a distinctive graphical signature.
| Type | Limit Exists? | Graphical Clue | Example |
|---|---|---|---|
| Removable | Yes | Open circle (hole) on the curve | f(x) = (x² − 4)/(x − 2) at x = 2 |
| Jump | No (two-sided) | Curve breaks; two separate pieces | Piecewise function changing rule at x = 1 |
| Infinite | No (outputs unbounded) | Vertical asymptote | f(x) = 1/(x − 3) at x = 3 |
Worked Example — Limits in a Modelling Context
A biologist models the concentration C(t) of a nutrient in a lake (in mg/L) after t hours using the function:
Comparing the Graphical and Numerical Methods
Both the graphical and numerical approaches give you a way to estimate limits, but each has strengths and weaknesses. Understanding when to use which method will make you more efficient on IB assessments and more confident in real-world modelling.
| Criterion | Graphical Method | Numerical Method |
|---|---|---|
| Speed | Very fast — a glance at the graph often suffices | Slower — you must compute several function values |
| Precision | Limited by graph resolution; hard to read exact values | High — you can use as many decimal places as you like |
| One-sided limits | Easy to see — trace the curve from each side | Also straightforward — make two columns (left and right) |
| Best when… | You have a GDC or graph-plotting tool, or need a quick qualitative answer | You need a quantitative estimate and can compute or use a calculator |
| Limitations | Can be misleading if the graph's scale hides behaviour near the point | Gives an estimate, not a proof — rounding issues can sometimes mislead |
Connection to Derivatives and Further Calculus
Limits are not just a stand-alone topic — they are the foundation on which the rest of calculus is built. In the next part of the IB course (SL 5.1 and beyond), you will use limits to define the derivative, which measures instantaneous rates of change. The derivative of f(x) is defined as the limit of the difference quotient: lim(h→0) [f(x + h) − f(x)] / h. Without understanding limits, that definition would make no sense.
| Concept | This Lesson (Intro) | Coming Next (Derivatives) |
|---|---|---|
| What we find | The value a function approaches at a point | The slope (rate of change) of a function at a point |
| Notation | lim(x→a) f(x) | f′(x) = lim(h→0) [f(x+h) − f(x)] / h |
| Methods | Graphical, numerical, algebraic simplification | Limit definition, then shortcut rules (power rule, etc.) |
| Application | Predicting function behaviour; modelling continuity | Finding velocity, optimising quantities, tangent lines |
Continuity will also remain important: many modelling scenarios require functions that are continuous over an interval (for instance, population models or temperature curves). A solid grasp of limits and continuity from this lesson will pay dividends throughout the rest of your IB calculus work.
Practice Problems
Lesson Summary
A limit describes the value a function approaches as the input nears a target, even when the function cannot be evaluated there directly. You can estimate limits using the graphical method (tracing the curve from both sides on a graph) or the numerical method (building a table of values that creep closer to the target). The two-sided limit exists only when the left-hand limit and right-hand limit agree.
A function is continuous at a point when three conditions are satisfied: the function is defined there, the limit exists, and the limit equals the function value. Failures of continuity are classified as removable (holes), jump, or infinite discontinuities. These foundational ideas prepare you for the definition of the derivative and the broader world of calculus-based modelling.