Limits & Continuity — SL 5.1 Limits and continuity in modelling (intro; graphical and numerical) - IB Mathematics Applications and Interpretation

SECTION 1

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.

c. 250 BCE • Archimedes & Exhaustion

Archimedes used the method of exhaustion to approximate the area of a circle by inscribing polygons with increasingly many sides — an early, intuitive use of limits.

1670s • Newton & Leibniz Invent Calculus

Isaac Newton and Gottfried Leibniz independently developed calculus. Newton spoke of 'ultimate ratios' — essentially limits — to describe instantaneous rates of change.

1821 • Cauchy Formalises Limits

Augustin-Louis Cauchy gave the first rigorous definition of a limit, laying the groundwork for the epsilon-delta framework still used in university courses today.

Present • Limits in Modelling

Today, limits underpin everything from predicting population growth to analysing sensor data. In the IB Applications & Interpretation course, you meet limits as the gateway to differential calculus and real-world modelling.

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.

SECTION 2

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

We say the limit of f(x) as x approaches a equals L if f(x) gets arbitrarily close to L as x gets close to a from both sides. Written: lim(x→a) f(x) = L.

One-Sided Limits

The left-hand limit (x → a⁻) considers values less than a, while the right-hand limit (x → a⁺) considers values greater than a. The two-sided limit exists only when both one-sided limits agree.

Continuity

A function is continuous at x = a when three conditions hold: f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a). Informally, you can draw the graph through that point without lifting your pen.

Discontinuity

A discontinuity occurs when any of the three continuity conditions fails. Common types include holes (removable), jumps, and vertical asymptotes (infinite discontinuities).

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KEY TAKEAWAY

Think of a limit like using GPS to predict your destination. Even if the road is blocked right at the address (the function is undefined there), the GPS can still tell you exactly where you were headed based on the direction you were travelling. A limit describes the intended destination of a function, not whether the function actually arrives.

SECTION 3

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.

Left panel: The curve passes smoothly through x = 2, so the limit equals f(2) and the function is continuous. Centre panel: There is a hole at x = 2 (open circle), but the limit still exists because both sides approach the same y-value. The filled dot shows f(2) is defined elsewhere. Right panel: The left-hand and right-hand limits disagree, so the two-sided limit does not exist — this is a jump discontinuity.

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.

SECTION 4

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.

LIMIT NOTATION

lim f(x) = L x→a

Read as: 'the limit of f(x) as x approaches a equals L.' Here a is the input value being approached, L is the output value the function approaches, and f(x) is the function rule.

ONE-SIDED LIMITS

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

The superscript ⁻ means we approach a from values less than a; the superscript ⁺ means we approach from values greater than a. The two-sided limit exists only when L⁻ = L⁺.

CONTINUITY TEST

f is continuous at x = a ⟺ f(a) is defined AND lim f(x) exists AND lim f(x) = f(a) x→a x→a

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

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

Numerical estimation of lim(x→2) (x² − 4)/(x − 2). Both sides approach 4.

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.

SECTION 5

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.

Removable: The limit exists but the function value is either missing or doesn't match. Graphically, you see an open circle (hole). Jump: The left-hand and right-hand limits are different finite values. Infinite: The function grows without bound near the point, indicated by a vertical asymptote.

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

SECTION 6

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:

CONCENTRATION MODEL

C(t) = (5t² − 20)/(t² − 4)

Note that when t = 2 the denominator equals zero, so C(2) is undefined. The biologist wants to know the expected concentration at exactly t = 2 hours.

Finding lim(t→2) C(t) — Graphically and Numerically

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Step 1 — Identify the Problem

Direct substitution gives C(2) = (5(4) − 20)/(4 − 4) = 0/0, which is undefined. The 0/0 form tells us we likely have a removable discontinuity and the limit may still exist.
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Step 2 — Numerical Approach (Table)

Build a table with t values approaching 2 from both sides. For t = 1.9: C(1.9) = (5(3.61) − 20)/(3.61 − 4) = (18.05 − 20)/(−0.39) = (−1.95)/(−0.39) = 5. For t = 1.99: C(1.99) ≈ 5. For t = 2.01: C(2.01) ≈ 5. The outputs all cluster around 5.

Numerical estimate: limit ≈ 5 mg/L
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Step 3 — Algebraic Confirmation

Factor the expression: C(t) = 5(t² − 4)/(t² − 4) = 5(t − 2)(t + 2)/((t − 2)(t + 2)). Cancel the common factor (t − 2)(t + 2) — valid when t ≠ 2 — to get C(t) = 5 for all t ≠ 2.

lim(t→2) C(t) = 5
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Step 4 — Interpret in Context

The model predicts that the nutrient concentration approaches 5 mg/L as time nears the 2-hour mark, even though the formula itself cannot be evaluated at exactly t = 2. The discontinuity is removable: if the biologist redefines C(2) = 5, the model becomes continuous.

SECTION 7

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

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KEY TAKEAWAY

Think of the graphical method as looking at a map from above — great for seeing the big picture and the overall direction. The numerical method is more like checking the odometer at closer and closer intervals — it gives you precise readings. In practice, the best strategy is to use both methods together: start with a graph for intuition, then confirm with a table of values.

SECTION 8

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain, in your own words, why lim(x→a) f(x) can exist even when f(a) is undefined. Give a real-life analogy to support your explanation.

Show answer

A limit describes the value a function approaches, not the value it actually reaches. Consider a train scheduled to arrive at a station at noon: even if the station is closed and the train cannot stop, the train was still heading towards that station — the 'intended destination' is noon at that station. Similarly, a function may have a hole at x = a (so f(a) is undefined), yet the outputs on both sides of a may converge to a single value L. That convergence is the limit.

Question 2: BASIC CALCULATION

Use a table of values to estimate lim(x→3) (x² − 9)/(x − 3). Show at least three values from each side.

Show answer

From the left: f(2.9) = (8.41 − 9)/(−0.1) = 5.9; f(2.99) = 5.99; f(2.999) = 5.999. From the right: f(3.1) = (9.61 − 9)/0.1 = 6.1; f(3.01) = 6.01; f(3.001) = 6.001. Both sides approach 6, so the limit is 6. This can be confirmed algebraically: (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3, and at x = 3 this equals 6.

Question 3: INTERMEDIATE

A piecewise function is defined as: f(x) = 2x + 1 for x < 4, and f(x) = 3x − 3 for x ≥ 4. Determine whether lim(x→4) f(x) exists and whether f is continuous at x = 4.

Show answer

From the left: lim(x→4⁻) (2x + 1) = 2(4) + 1 = 9. From the right: lim(x→4⁺) (3x − 3) = 3(4) − 3 = 9. Since both one-sided limits equal 9, the two-sided limit exists and equals 9. Now check continuity: f(4) = 3(4) − 3 = 9, which matches the limit. All three conditions for continuity are met, so f is continuous at x = 4.

Question 4: APPLIED

A phone battery's charge level (in %) is modelled by B(t) = (100t)/(t + 1) where t is time in hours since charging began. Use numerical or graphical methods to determine lim(t→∞) B(t) and explain what this means in context.

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As t grows large, evaluate: B(10) ≈ 90.9%, B(100) ≈ 99.0%, B(1000) ≈ 99.9%. The outputs approach 100. You can also rewrite B(t) = 100/(1 + 1/t); as t → ∞, the term 1/t → 0, so B(t) → 100. In context, the battery charge approaches but never quite reaches 100%. This is a common feature of asymptotic growth models — the function has a horizontal asymptote at y = 100.

Question 5: CRITICAL THINKING

A student claims: 'If f(a) = 5, then lim(x→a) f(x) must also equal 5.' Is this claim always true? Construct a counterexample using a piecewise function and explain how it disproves the claim.

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The claim is not always true. Consider the piecewise function: f(x) = x + 2 for x ≠ 3, and f(3) = 5. At x = 3, f(3) = 5 as stated. However, lim(x→3) f(x) = lim(x→3) (x + 2) = 5. In this particular case they happen to agree, so let's refine: define g(x) = x + 2 for x ≠ 3, and g(3) = 5. Then lim(x→3) g(x) = 5. For a true counterexample, use g(x) = x + 2 for x ≠ 3 and g(3) = 10. Now g(3) = 10, but lim(x→3) g(x) = 5 ≠ 10. The function value and the limit are independent — f(a) tells you the actual output at x = a, while the limit only cares about what happens near x = a. Continuity requires both to exist and be equal; without that, f(a) and the limit can differ.

SUMMARY

Lesson Summary

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.