📌 1. Understanding Limits
- A limit describes the value that a function approaches as the input variable gets closer to a particular value.
- Limits focus on behaviour near a point, not necessarily the function’s value at that point.
- A function may have a limit at x = a even if f(a) is undefined or different from the limit value.
- Limits allow mathematicians to describe smooth change and form the foundation of calculus.
- They are essential for defining continuity and derivatives rigorously.
🌍 Real-World Connection
Limits model real phenomena such as instantaneous speed, where average speed over smaller and smaller time intervals approaches a single value.
📌 2. One-Sided Limits & Existence of Limits
- A left-hand limit describes function behaviour as x approaches a value from smaller inputs.
- A right-hand limit describes function behaviour as x approaches a value from larger inputs.
- A two-sided limit exists only if both one-sided limits exist and are equal.
- If one-sided limits differ, the function has a jump discontinuity.
- Checking one-sided limits is essential in piecewise functions and graph-based questions.
🧠 Examiner Tip
- Always explicitly state both one-sided limits before concluding that a limit exists.
- Writing “does not exist” without justification loses method marks.
📌 3. Continuity & Types of Discontinuities
- A function is continuous at x = a if the limit exists and equals the function value.
- A removable discontinuity occurs when the limit exists but the function value is missing or incorrect.
- A jump discontinuity occurs when left and right limits exist but are not equal.
- An infinite discontinuity occurs when the function approaches infinity near a vertical asymptote.
- Continuity is a prerequisite for differentiability.
📐 IA Spotlight
In the IA, continuity can be used to justify model validity. Students should explain why a model behaves smoothly near important threshold values.
📌 4. Derivative as a Limit
- The derivative is defined as the limit of the average rate of change.
- It measures how fast one variable changes relative to another at an instant.
- Geometrically, the derivative represents the gradient of the tangent line.
- Physically, it represents quantities such as velocity and acceleration.
- The derivative connects algebraic expressions to graphical behaviour.
f′(a) = limh→0 [f(a+h) − f(a)] / h
🔍 TOK Perspective
Is it valid to describe real-world change using infinitely small quantities, or does this idealisation distort reality?
📌 PRACTICE QUESTIONS
🧠 Multiple Choice Questions
MCQ 1. What does limx→3 f(x) = 5 mean?
A. f(3) = 5
B. f(x) equals 5 near x = 3
C. f(x) approaches 5 as x approaches 3
D. The function is continuous at x = 3
Answer: C
The definition of a limit concerns approach behaviour, not function value or continuity.
MCQ 2. A limit does not exist if:
A. The function is undefined at the point
B. One-sided limits are unequal
C. The graph has a hole
D. The function is continuous
Answer: B
Unequal one-sided limits indicate a jump discontinuity.
MCQ 3. Which situation guarantees continuity at x = a?
A. f(a) exists
B. The limit exists
C. lim f(x) = f(a)
D. The function is defined nearby
Answer: C
Continuity requires both existence of the limit and equality with the function value.
MCQ 4. The derivative represents:
A. Average rate of change
B. Area under a curve
C. Instantaneous rate of change
D. Total change
Answer: C
✏️ Short Answer Questions
SAQ 1. Explain why a function can have a limit at a point where it is undefined.
A limit describes approach behaviour. Even if f(a) is undefined, values of f(x) near a can approach a single number.
SAQ 2. State two interpretations of the derivative.
The derivative represents the gradient of the tangent line and the instantaneous rate of change.—
Long Answer / Explainer Questions
Q7.
The function f is defined by
f(x) = (x² − 4)/(x − 2).
(a) Investigate the behaviour of f(x) as x approaches 2 using algebraic methods.
(b) State whether the limit of f(x) as x approaches 2 exists and justify your answer.
(c) Explain why f is not continuous at x = 2, even though the limit exists.
(d) Describe how the function could be redefined to make it continuous at x = 2.
Full Worked Solution:
(a) Algebraic investigation of the limit
We begin by factoring the numerator:
x² − 4 = (x − 2)(x + 2)
Substituting into the expression for f(x):
f(x) = (x − 2)(x + 2) / (x − 2)
For all x ≠ 2, the factor (x − 2) cancels, giving:
f(x) = x + 2
(b) Existence of the limit
Since f(x) simplifies to x + 2 for all values of x except 2, we evaluate the limit by substitution:
limx→2 f(x) = 2 + 2 = 4
Therefore, the limit exists and is equal to 4.
(c) Continuity at x = 2
Although the limit exists, the original function f(x) is undefined at x = 2 because the denominator becomes zero.
Since continuity requires:
- the function value to exist, and
- the function value to equal the limit,
f is not continuous at x = 2.
(d) Redefinition for continuity
To make the function continuous at x = 2, define:
f(2) = 4
This fills the removable discontinuity and ensures that the function value matches the limit.
Conclusion:
This example illustrates a removable discontinuity, where algebraic simplification reveals the underlying behaviour of the function near the point.
Q8.
The displacement s of a particle moving in a straight line is given by s(t) = t² − 3t, where t is measured in seconds.
(a) Explain how the concept of a limit can be used to define the instantaneous velocity of the particle at time t.
(b) Use first principles to find the instantaneous velocity of the particle at t = 2.
(c) Interpret the meaning of your result in the context of motion.
(d) Comment on the relationship between average velocity and instantaneous velocity.
Full Worked Solution:
(a) Limit-based definition of velocity
Instantaneous velocity cannot be measured directly because it refers to motion at a single instant.
Instead, we calculate the average velocity over a small time interval h:
Average velocity = [s(t + h) − s(t)] / h
The instantaneous velocity is defined as the limit of this average velocity as h approaches 0.
(b) First-principles calculation
Given s(t) = t² − 3t, compute s(2 + h):
s(2 + h) = (2 + h)² − 3(2 + h)
= 4 + 4h + h² − 6 − 3h
= h² + h − 2
Now compute the difference quotient:
[s(2 + h) − s(2)] / h
= [(h² + h − 2) − (−2)] / h
= (h² + h) / h
= h + 1
Taking the limit as h approaches 0:
limh→0 (h + 1) = 1
Thus, the instantaneous velocity at t = 2 is 1 m/s.
(c) Interpretation
This result means that at exactly 2 seconds, the particle is moving forward at a speed of 1 metre per second.
It represents the slope of the tangent to the displacement–time graph at t = 2.
(d) Average vs instantaneous velocity
Average velocity depends on the chosen time interval, whereas instantaneous velocity is a single, well-defined value obtained by letting the interval shrink to zero.
Limits allow this transition from discrete measurement to continuous description.
Conclusion:
This question demonstrates how limits form the conceptual bridge between algebraic functions and physical motion.