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limits

Why the Limit Does Not Care About the Function Value

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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:

\lim_{x\to 2}f(x)=5

and

f(2)=100

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

\lim_{x\to 2}f(x)=5

The fact that:

f(2)=100

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:

\lim_{x\to a}f(x)=f(a)

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.

\text{Limit} \rightarrow \text{near the point} \text{Function value} \rightarrow \text{at the point}

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

Limit Value vs Function Value

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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:

\lim_{x\to 1}f(x)=3

but

f(1)=0

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:

\lim_{x\to a}f(x)\neq f(a)\text{ in general}

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