Limits & Continuity — SL 5.1 Limits and continuity (intro; graphical and numerical) - IB Mathematics Analysis and Approaches

SECTION 1

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.

~450 BCE • Zeno's Paradoxes

The Greek philosopher Zeno poses paradoxes about infinite processes, hinting at the need for a concept of limits to resolve them.

~250 BCE • Archimedes' Method of Exhaustion

Archimedes approximates the area of a circle by inscribing polygons with more and more sides — an early use of limit-like reasoning.

1684 • Leibniz & Newton Develop Calculus

Both Newton and Leibniz independently create calculus, using informal notions of infinitely small quantities to define derivatives and integrals.

1821 • Cauchy Formalises Limits

Augustin-Louis Cauchy provides the first precise definition of a limit, replacing vague ideas of 'infinitely close' with rigorous inequalities.

1861 • Weierstrass and the ε–δ Definition

Karl Weierstrass refines Cauchy's work into the modern epsilon-delta definition, giving calculus the logical foundation it uses today.

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.

SECTION 2

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

The value that f(x) gets closer and closer to as x approaches a particular number c. We write lim f(x) as x → c. The function does not need to equal the limit at c — it only needs to approach it.

Left-Hand & Right-Hand Limits

The left-hand limit examines values of x approaching c from below (x → c⁻). The right-hand limit examines x approaching c from above (x → c⁺). A two-sided limit exists only when both one-sided limits agree.

Continuity at a Point

A function f is continuous at x = c if three conditions hold: (1) f(c) is defined, (2) lim f(x) as x → c exists, and (3) lim f(x) = f(c). If any condition fails, there is a discontinuity.

Types of Discontinuity

A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(c). A jump discontinuity occurs when the one-sided limits differ. An infinite discontinuity occurs at a vertical asymptote.

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KEY TAKEAWAY

Think of a limit like using GPS navigation: you care about the road leading toward your destination, not whether you actually arrive. If both the road from the left and the road from the right point to the same place, the limit exists — even if there's a pothole (hole in the graph) right at that spot. Continuity means the road is smooth, the destination exists, and you actually arrive exactly where the roads point.

SECTION 3

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.

Three scenarios at different x-values. A (cyan): a removable discontinuity — the limit is 2 but f(1) = 4 (filled dot above the hole). B (violet): a jump discontinuity — the left-hand and right-hand limits disagree, so no two-sided limit exists. C (pink): continuity — the curve passes smoothly through the point, and the limit equals the function value.

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.

SECTION 4

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.

LIMIT NOTATION

lim f(x) = L x → c

Read as: 'the limit of f(x) as x approaches c equals L.' Here, c is the x-value being approached, L is the y-value the function gets arbitrarily close to, and f(x) is the function rule.

ONE-SIDED LIMITS

lim f(x) = L⁻ (left-hand) lim f(x) = L⁺ (right-hand) x → c⁻ x → c⁺

The superscript minus sign means x approaches c from values less than c; the plus sign means from values greater than c. The two-sided limit exists if and only if L⁻ = L⁺.

CONTINUITY CONDITIONS

f is continuous at x = c ⟺ (1) f(c) is defined, (2) lim f(x) exists, (3) lim f(x) = f(c) x→c x→c

All three conditions must hold simultaneously. If even one fails, the function is discontinuous at that point.

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.

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IB Exam Tip

On Paper 1 (no calculator), the IB often tests limits by giving a piecewise function and asking whether the function is continuous at the boundary. Always check all three continuity conditions and state each one explicitly for full marks.

SECTION 5

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

Numerical investigation of lim (x² − 4)/(x − 2) as x → 2

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.

The graph of f(x) = (x² − 4)/(x − 2) is the straight line y = x + 2 with a hole at x = 2. The cyan arrow shows values approaching from the left, the green arrow from the right — both converge to y = 4.

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How Many Values Are Enough?

For IB problems, using 4–5 x-values from each side is typically sufficient to establish a clear trend. Choose values that get progressively closer to c: for instance, 1.9, 1.99, 1.999, 1.9999 when approaching 2 from the left.

SECTION 6

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.

Piecewise Function: Limits & Continuity

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Step 1 — Read the Problem

Consider the function f defined by: f(x) = x² + 1 for x < 2, and f(x) = 3x − 1 for x ≥ 2. Determine (a) the left-hand limit as x → 2, (b) the right-hand limit as x → 2, (c) whether the overall limit exists, and (d) whether f is continuous at x = 2.
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Step 2 — Find the Left-Hand Limit

For x < 2, the rule is f(x) = x² + 1. As x approaches 2 from the left, we substitute x = 2 into this expression (since it's a polynomial, direct substitution works): lim x→2⁻ f(x) = (2)² + 1 = 4 + 1.

Left-hand limit = 5
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Step 3 — Find the Right-Hand Limit

For x ≥ 2, the rule is f(x) = 3x − 1. As x approaches 2 from the right, we substitute x = 2: lim x→2⁺ f(x) = 3(2) − 1 = 6 − 1.

Right-hand limit = 5
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Step 4 — Does the Two-Sided Limit Exist?

Because the left-hand limit (5) equals the right-hand limit (5), the two-sided limit exists.

lim x→2 f(x) = 5
5
Step 5 — Check Continuity

We verify the three conditions: (1) f(2) is defined — using the x ≥ 2 piece, f(2) = 3(2) − 1 = 5. ✓ (2) The limit as x → 2 exists and equals 5. ✓ (3) lim x→2 f(x) = f(2), since both equal 5. ✓

f is continuous at x = 2.

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Numerical Check

You can verify with a quick table: f(1.99) = 1.99² + 1 = 4.9601, f(1.999) = 4.996001, f(2.01) = 3(2.01) − 1 = 5.03, f(2.001) = 5.003. Both sides converge to 5, confirming the algebraic result.

SECTION 7

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

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KEY TAKEAWAY

Think of the graphical method as looking at a satellite photo of a road, and the numerical method as reading GPS coordinates along that road. The photo gives you the big picture instantly — is there a bridge, a dead end, or a fork? The coordinates give you exact locations. In the IB course, the best strategy is to combine both: use the graph to understand the shape of the behaviour, and use the table to nail down the number.

SECTION 8

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain in your own words why the limit of a function as x → c can exist even if f(c) is undefined. Give one example of a function where this occurs.

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A limit describes the value that f(x) approaches as x gets close to c, not the value at c itself. The function doesn't need to be defined at c for a trend to exist in the surrounding values. For example, f(x) = (x² − 9)/(x − 3) is undefined at x = 3, but by factoring we get (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3. As x → 3, the simplified expression approaches 6, so the limit is 6 even though f(3) doesn't exist. The graph shows a straight line y = x + 3 with a hole at (3, 6).

Question 2: BASIC CALCULATION

Find lim x→−1 (x² + 3x + 2)/(x + 1) by factoring the numerator.

Show answer

Direct substitution gives 0/0, so we factor: x² + 3x + 2 = (x + 1)(x + 2). The expression becomes (x + 1)(x + 2)/(x + 1) = x + 2 for x ≠ −1. Substituting x = −1 into x + 2 gives −1 + 2 = 1. Therefore the limit is 1. This is a removable discontinuity — the graph of the original function is the line y = x + 2 with a hole at (−1, 1).

Question 3: INTERMEDIATE

A piecewise function is defined as g(x) = 2x + k for x < 3, and g(x) = x² for x ≥ 3. Find the value of k that makes g continuous at x = 3.

Show answer

For g to be continuous at x = 3, we need the left-hand limit to equal the right-hand limit and both to equal g(3). The right-hand limit and function value come from x²: g(3) = 9. The left-hand limit comes from 2x + k: as x → 3⁻, this gives 2(3) + k = 6 + k. Setting 6 + k = 9 gives k = 3. You can verify: with k = 3, the left-hand expression at x = 3 gives 2(3) + 3 = 9, which matches g(3) = 9, so all three continuity conditions are satisfied.

Question 4: APPLIED

A car's position along a straight road is given by s(t) = t² + 2t metres, where t is in seconds. Using a table of values, estimate the limit of [s(t) − s(1)]/(t − 1) as t → 1 from both sides. What does this limit represent physically?

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First, s(1) = 1 + 2 = 3. Now build the table: at t = 0.9, [s(0.9) − 3]/(0.9 − 1) = [0.81 + 1.8 − 3]/(−0.1) = (−0.39)/(−0.1) = 3.9. At t = 0.99, the value is 3.99. At t = 1.01, we get [1.0201 + 2.02 − 3]/0.01 = 0.0401/0.01 = 4.01. At t = 1.1, the value is 4.1. Both sides converge to 4, so the limit is 4. Physically, this limit represents the car's instantaneous velocity at t = 1 second, which is 4 m/s. This is precisely how the derivative is defined — you're previewing SL 5.1's connection to differentiation.

Question 5: CRITICAL THINKING

Consider f(x) = sin(1/x). Investigate the limit as x → 0 using a table of values (try x = 0.1, 0.01, 0.001, 0.0001). Does the limit exist? Can a function fail to have a limit at a point even if the function is defined everywhere near that point? Explain your reasoning.

Show answer

Computing: f(0.1) = sin(10) ≈ −0.544, f(0.01) = sin(100) ≈ −0.506, f(0.001) = sin(1000) ≈ 0.827, f(0.0001) = sin(10000) ≈ −0.306. The values don't settle toward any single number — they keep oscillating. This happens because as x → 0, the input 1/x grows without bound, and sin oscillates forever between −1 and 1. The limit does not exist, even though f(x) is defined for every x ≠ 0. This shows that being defined near a point is necessary but not sufficient for a limit to exist; the function outputs must also converge to a single value. This is a fundamentally different type of 'no limit' from a jump discontinuity.

SUMMARY

Lesson Summary

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.