limits

Understanding Limits: Why the Limit Value Can Be Independent of the Function Value

June 19, 2026 by Splendid Leave a Comment

The Big Idea

One of the most important discoveries students make when studying calculus is that a limit value and a function value are not necessarily the same thing.

A function can approach one value while having a completely different value at the point itself.

This idea often feels strange at first because, in everyday arithmetic, a value is simply a value. Calculus introduces a new perspective: sometimes what happens near a point matters more than what happens at the point.

A Common Example

Consider the function:

$f(x)=\begin{cases}x^2, & x\neq 1\\100, & x=1\end{cases}$

For all values close to 1, the function behaves exactly like $x^2$ .

Therefore,

$\lim_{x\to1}f(x)=1$

However, the function value at the point is:

$f(1)=100$

Notice that:

The limit is 1.
The function value is 100.

Both statements are true simultaneously.

Why Does This Happen?

A limit examines the behavior of a function near a point.

When evaluating:

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

the values of $f(x)$ are considered for points very close to $a$ .

The actual value $f(a)$ is not part of the limit definition.

In other words, limits focus on the journey, not necessarily the destination.

Applying Limit Laws

Consider the information:

$\lim_{x\to1}f(x)=3$

and

$f(1)=0$

Suppose it is also known that:

$\lim_{x\to1}g(x)=7$

Now evaluate:

$\lim_{x\to1}[2f(x)+g(x)]^2$

Using the limit laws:

$\lim_{x\to1}2f(x)+g(x)]^2=\left(2\lim_{x\to1}f(x)+\lim_{x\to1}g(x)\right)^2$

Substituting the known limits:

$\lim_{x\to1}[2f(x)+g(x)]^2=\left(2(3)+7\right)^2=13^2=169$

Notice that the value:

$f(1)=0$

plays no role in the calculation because limits depend on the behavior of the function near the point rather than the value at the point itself.

Can Exceptional Values Be Ignored?

This observation leads to an important idea in calculus.

Suppose a function is continuous almost everywhere except for a few isolated points where different values have been assigned.

In many calculus applications, the limit values provide a better description of the function’s overall behavior than those isolated exceptions.

For example:

$f(x)=\begin{cases}x^2, & x\neq 1\\100, & x=1\end{cases}$

The graph behaves like $x^2$ everywhere except at one point.

A mathematician would often say that the function is “essentially” behaving like $x^2$ and that the point at $x=1$ represents a removable defect.

Removable Discontinuities

A discontinuity is called removable when the limit exists but the function value differs from that limit.

In the previous example:

$\lim_{x\to1}f(x)=1$

but

$f(1)=100$

The discontinuity can be removed by redefining the function value:

$f(1)=1$

After doing so, the function becomes continuous.

Why Calculus Prefers Limits

Calculus focuses on:

Continuity
Rates of change
Derivatives
Areas under curves
Long-term behavior

In many of these situations, changing a function at a single point has no effect on the overall result.

For example, the area under a curve remains unchanged whether:

$f(1)=1$

$f(1)=1000000$

A single point contributes no area.

This is why limits play such a central role in calculus.

A Helpful Mental Model

A useful way to think about functions is to imagine two layers:

Layer 1: Overall Behavior

How the graph behaves as values move through the domain.

Layer 2: Exceptional Point Values

Special values assigned at specific points.

Limits study Layer 1.

Function values study Layer 2.

When only a few exceptional points exist, calculus is often more interested in the overall behavior represented by the limits.

Key Takeaways

A limit value and a function value are different concepts.
The limit depends on values near a point.
The function value depends on the exact point.
A limit can exist even when the function value is different.
Isolated exceptional values often do not affect the overall behavior of a function.
Removable discontinuities occur when a limit exists but the function value does not match the limit.

Most importantly:

$\lim_{x\to a}f(x)\neq f(a)$

in general.

They are equal only when the function is continuous at the point.

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

Understanding this distinction is one of the foundational ideas that makes calculus possible.

Why the Limit Does Not Care About the Function Value

June 16, 2026 by Splendid Leave a Comment

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:

Look at values near the point.
Determine what the function approaches.
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

June 16, 2026 by Splendid Leave a Comment

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).

Understanding Limits: Why the Limit Value Can Be Independent of the Function Value

The Big Idea

A Common Example

Why Does This Happen?

Applying Limit Laws

Can Exceptional Values Be Ignored?

Removable Discontinuities

Why Calculus Prefers Limits