Common Misconception

Many students think that to find a limit, we simply use the function value.

This is not always true.

A limit looks at what happens near a point, not necessarily at the point.

Example

Suppose:

and

When evaluating the limit, we use the value the function is approaching:

The fact that:

does not change the limit.

Real-Life Analogy

Think of approaching a destination.

• The limit is the place you are heading toward.
• The function value is where you are standing at the exact moment.

You can be approaching one location while standing somewhere else.

Exam Tip

When solving limit problems:

1. Look at values near the point.
2. Determine what the function approaches.
3. Do not automatically substitute the function value.

Important Formula

A function is continuous only when:

If this equality does not hold, the function is not continuous at (a).

Final Takeaway

The limit describes approaching behavior, while the function value describes actual behavior at a point.

That is why the limit value and the function value are independent of each other.