Limit Definition in Maths (Explained Simply for Students)

A limit is the value that a function or sequence gets closer and closer to, even if it never reaches it.

You can think of it like walking toward a door:

• You keep getting closer
  You might never touch it exactly
  ➡️ But the limit is the door.
  
  

In maths, limits help us understand what happens to a value as it approaches a certain point.

Why Are Limits Important?

Limits let us:

• Understand how functions behave

• Describe values we cannot plug directly into a function

• Prepare for concepts like continuity, slopes, and derivatives

• Build the foundation for calculus
  
  

Even if a point is missing on a graph, a limit tells you what value the function is approaching.

Easy Examples of Limits

✍🏼 Example 1 — A Number Sequence

Sequence:
2.9, 2.99, 2.999, 2.9999…

This gets closer to 3.
So the limit is:

lim = 3

✍🏼 Example 2 — Filling a Cup

Imagine filling a cup with water.
You pour slowly and the amount rises:

80% → 90% → 95% → 99% → 99.9%

It gets closer to 100%, even if you stop before filling exactly to the top.

The limit = 100%

✍🏼 Example 3 — A Graph With a Missing Point

Even if a point is missing on the graph, the limit tells us the value the graph is about to reach.

Left-Hand and Right-Hand Limits (Simple)

• Left-hand limit: approaching a value from the left

• Right-hand limit: approaching a value from the right
  
  

If both sides match → the limit exists.

Basic Limit Rules (Easy Version)

1. Constant rule:
   lim(c) = c
   (The limit of a constant is the constant.)
   
   

2. Addition rule:
   lim(f + g) = lim(f) + lim(g)
   
   

3. Multiplication rule:
   lim(f × g) = lim(f) × lim(g)
   
   

4. Direct substitution:
   If a function is continuous, you can just plug in the value.
   
   

Practice Exercises (With Answers)

Try these simple problems:

Questions

1. lim x→3 (x + 2)

2. lim x→5 (2x)

3. lim x→2 (x²)

4. Sequence approaches: 4.1, 4.01, 4.001 → ?

5. A line approaches height 10 → what is the limit?
   
   

Answers

1. 5

2. 10

3. 4

4. 4

5. 10

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