Limit Value vs Function Value

The Big Idea

A limit value and a function value are two different concepts.

• The limit tells us what value the function is approaching.
• The function value tells us the actual value of the function at a specific point.

Therefore, the limit can exist even if the function value is different or does not exist.

Key Statement

The limit value is independent of the function value, and the function value is independent of the limit value.

Example

Suppose:

but

This means:

• As (x) gets close to 1, the function values get close to 3.
• At exactly (x=1), the function’s value is 0.

Both statements can be true at the same time.

Visual Interpretation

Imagine a graph with:

• An open circle at ((1,3))
• A filled dot at ((1,0))

The graph approaches 3, so the limit is 3.

The filled dot shows the actual function value, which is 0.

Takeaway

Always remember:

They are equal only when the function is continuous at (x=a).

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