Mathematics - Limits and Continuity

Mathematics Discovery Game

The Mystery of Limits

Discover what happens when a number approaches another number.

Your first discovery 1 / 8

SLIDE 01 - THE MYSTERY

Can you get closer without reaching?

Imagine you are standing 10 metres away from a wall. You take a step forward. Then another. Then another.

A simple approaching experiment

Think about it. What happens to your distance from the wall as you keep moving closer?

SLIDE 02 - OBSERVATION

Watch the distance become smaller

Let us make the experiment more interesting. You can control how far the person is from the wall.

Move the person towards the wall

Distance from the wall 10.0 metres

Move the slider towards the left to get closer.

Observe carefully. Does the distance have to become exactly zero before we can say that it is approaching zero?

SLIDE 03 - MATHEMATICAL DISCOVERY

Now replace the wall with a number

Mathematics uses the same idea. Instead of moving towards a wall, we let a variable called x move towards a particular number.

x approaches 2

Imagine a mathematical machine. It receives x as input and produces x squared as output.

The function machine

Change the input x 2.00

Input x 2.00

Operation x squared

Output 4.00

Discover the relationship. When x gets closer to 2, what happens to x squared?

SLIDE 04 - INTERACTIVE GRAPH

See the limit on a graph

Now look at the graph of the function f(x) = x squared. The target input is x = 2.

Function graph: f(x) = x squared

Distance from x = 2 1.00

Current x 1.00

Current f(x) 1.00

Target output 4.00

What should you notice? As the input gets closer to 2 from either side, the output gets closer to 4.

What value is the graph approaching at x = 2?

SLIDE 05 - NUMERICAL INVESTIGATION

Look closely at the numbers

Let us calculate the output for inputs that are getting closer and closer to 2.

Approaching from the left

x	x squared
1.0	1.00
1.5	2.25
1.9	3.61
1.99	3.9601
1.999	3.996001

Approaching from the right

x	x squared
3.0	9.00
2.5	6.25
2.1	4.41
2.01	4.0401
2.001	4.004001

Discovery: The outputs from both sides are getting closer to 4. This is the numerical meaning of a limit.

Notice that the values are not necessarily equal to 4. They are approaching 4.

Which statement is correct?

SLIDE 06 - IMPORTANT DISTINCTION

Approaching is not the same as reaching

This is one of the most important ideas in the entire chapter.

Compare these two ideas

lim x squared = 4
x approaches 2

This statement tells us that the output approaches 4. It does not simply describe what happens at the exact input x = 2.

Can a function have a limit at x = 2 even if its value at x = 2 is not defined?

This idea will become very important when we study holes in graphs and the indeterminate form 0/0.

SLIDE 07 - YOUR FIRST CHALLENGE

Can you solve it yourself?

A function is defined as:

f(x) = 3x + 2

Your task is to find what value the function approaches when x approaches 2.

Think before calculating. What will be the value of the output near x = 2?

SLIDE 08 - DISCOVERY COMPLETE

You have discovered the meaning of a limit

You started with an everyday situation and reached a mathematical concept.

The central idea A limit tells us the value that a function approaches when its input approaches a particular number.

lim f(x) = L
x approaches a

Understand every symbol

f(x): The output of the function.

x: The input variable.

a: The target input value.

L: The value approached by the output.

lim: The mathematical instruction to investigate nearby behaviour.

First discovery unlocked

You now understand the basic meaning of a limit. The next investigation will explore what happens when we approach a number from the left and the right.

Next mission: Left-hand and Right-hand Limits

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Limits and Continuity / Stage 2

Left-hand and Right-hand Limits

Discover what happens when x approaches a number from two different directions.

Can you approach a point from two directions?

Imagine standing in front of a door marked with the number 2.

You can approach the door from the left side or from the right side.

From the left Numbers smaller than 2

1.9, 1.99, 1.999

From the right Numbers greater than 2

2.1, 2.01, 2.001

Both directions move towards 2. But they approach 2 from opposite sides.

Before moving ahead, think:

Is 1.99 smaller or greater than 2?

Experiment 1: Approach from the left

Move the slider. Watch x get closer to 2 from smaller values.

x = 2 - distance

1 2 3

Distance from 2 0.50

Current x value 1.50

Function output 6.50

f(x) = 3x + 2

As x gets closer to 2 from the left, the output gets closer to 8.

Left-hand approach means x is always smaller than the target value, but moves closer to it.

Experiment 2: Approach from the right

Now approach 2 from numbers that are greater than 2.

x = 2 + distance

1 2 3

Distance from 2 0.50

Current x value 2.50

Function output 9.50

f(x) = 3x + 2

As x gets closer to 2 from the right, the output also gets closer to 8.

Right-hand approach means x is always greater than the target value, but moves closer to it.

Compare both directions

We are studying the same function from two different sides.

f(x) = 3x + 2

From the left	From the right
1.9	2.1
1.99	2.01
1.999	2.001
7.7	8.3
7.97	8.03
7.997	8.003

Left output

Approaches 8

Right output

Approaches 8

The outputs approach the same number from both directions: 8.

What do you notice?

Mathematics gives these directions special names

We use different symbols to describe the two approaches.

lim _{x → 2⁻}  f(x)

Left-hand limit

The small minus sign means x approaches 2 from values smaller than 2.

lim _{x → 2⁺} f(x)

Right-hand limit

The small plus sign means x approaches 2 from values greater than 2.

Remember: the minus and plus signs show the direction of approach. They do not mean that we subtract or add something to the function.

See both directions on a graph

Consider the function f(x) = 3x + 2. Its graph is a straight line.

From left Move along the graph towards x = 2

From right Move along the graph towards x = 2

Both paths reach the same height as they approach x = 2. That height is 8.

The graph gives us a visual way to compare the left-hand and right-hand limits.

What if the two directions give different answers?

Now consider a new function:

f(x) = x + 2, when x is less than 2

f(x) = x + 4, when x is greater than 2

Left-hand limit

4

Right-hand limit

6

The two directions approach different values. The left-hand limit is 4, while the right-hand limit is 6.

Can the two-sided limit exist in this example?

Your discovery: when does a limit exist?

You have now seen two possible situations.

Situation A: Both sides approach the same value

Left = 8

Right = 8

The two-sided limit exists and equals 8.

Situation B: Both sides approach different values

Left = 4

Right = 6

The two-sided limit does not exist.

Left-hand limit = Right-hand limit

This equality is the condition for a two-sided limit to exist.

Important: this condition is about the values approached from both sides. It does not require the function to be defined at the target point.

Final challenge

If the left-hand limit is 5 and the right-hand limit is also 5, what is the two-sided limit?

Stage 2 completed

You can now distinguish left-hand limits from right-hand limits and decide whether a two-sided limit exists.

Next stage: Why does 0/0 create a problem?

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Limits and Continuity / Stage 3

Why is 0/0 indeterminate?

Discover why the same direct substitution can lead to completely different limits.

A strange result appears

Consider this function:

f(x) = (x² - 4) / (x - 2)

We want to find what happens when x approaches 2.

Try putting x = 2 directly into the expression.

(2² - 4) / (2 - 2)

0 / 0

We have reached a strange result. The numerator is zero, and the denominator is also zero.

Can we conclude that the limit is zero? Or does the limit not exist?

What do you think 0 / 0 means?

Look closer instead of substituting

When x is very close to 2, but not exactly 2, calculate the function.

f(x) = (x² - 4) / (x - 2)

x	Numerator	Denominator	Output
1.9	-0.39	-0.1	3.9
1.99	-0.0399	-0.01	3.99
1.999	-0.003999	-0.001	3.999
2.001	0.004001	0.001	4.001
2.01	0.0401	0.01	4.01
2.1	0.41	0.1	4.1

Notice something important: the output is getting closer to 4 from both sides.

Limit = 4

If direct substitution gave 0 / 0, why is the limit 4?

Experiment: move x towards 2

Use the slider to change x. Watch the numerator, denominator and output change.

f(x) = (x² - 4) / (x - 2)

Choose x 2.50

Numerator 2.25

Denominator 0.50

Output 4.50

Distance from 2 0.50

Move x closer to 2. You will see the numerator and denominator both approach zero, while the output approaches 4.

What happens when x is exactly 2?

Find the hidden pattern

There is a simpler way to understand the same function.

f(x) = (x² - 4) / (x - 2)

Remember the algebraic identity:

a² - b² = (a - b)(a + b)

So we can write:

x² - 4 = (x - 2)(x + 2)

f(x) = ((x - 2)(x + 2)) / (x - 2)

f(x) = x + 2

For x values other than 2, the common factor can be cancelled.

Important: cancellation does not make the original function defined at x = 2. It only gives an equivalent expression for x values other than 2.

Now when x approaches 2, the simplified expression approaches:

2 + 2 = 4

Can we use the simplified expression to find the limit?

Now look at a different function

Consider:

g(x) = (x - 2) / (x - 2)

Put x = 2 directly into the expression.

0 / 0

Again, we get 0 / 0. But when x is not equal to 2, we can simplify:

g(x) = 1

Try nearby values:

x	g(x)
1.9	1
1.99	1
2.01	1
2.1	1

The limit is 1, even though direct substitution gives 0 / 0.

What does this tell us?

One more function. A very different result.

Consider:

h(x) = (x - 2)² / (x - 2)

Direct substitution again gives:

0 / 0

But simplify the expression for x not equal to 2:

h(x) = x - 2

As x approaches 2, what does x - 2 approach?

Function 1 4 (x² - 4) / (x - 2)

Function 2 1 (x - 2) / (x - 2)

Function 3 0 (x - 2)² / (x - 2)

All three functions produce 0 / 0 when x = 2 is substituted directly. Yet their limits are 4, 1 and 0.

Which conclusion is correct?

See the missing point on a graph

Look at the graph of:

f(x) = (x² - 4) / (x - 2)

For x not equal to 2, this is the same as y = x + 2.

The line approaches the point (2, 4), but the original function has no value at x = 2 because its denominator becomes zero.

This missing point is often called a hole in the graph.

Does a missing point always mean the limit does not exist?

The big discovery

You have investigated three functions. All three gave the same result after direct substitution.

0 / 0

Function	Limit as x approaches 2
(x2 - 4) / (x - 2)	4
(x - 2) / (x - 2)	1
(x - 2)2 / (x - 2)	0

This is why 0 / 0 is called an indeterminate form. It does not determine the limit by itself.

What should you do when you get 0 / 0?

1. Do not assume the limit is zero.
2. Try factorisation and simplification.
3. Check values close to the target.
4. Study the graph or use another suitable method.

Final challenge

A function gives 0 / 0 when x = 3 is substituted. What can you conclude immediately?

Stage 3 completed

You now understand why 0 / 0 is indeterminate and why limits must be investigated using nearby values or algebraic methods.

Next stage: Continuity - what happens when a graph has a break, a hole or a jump?

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