Differential Calculus

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.

Understanding Derivatives, Relative Extrema, and Points of Inflection

June 16, 2026 by Splendid Leave a Comment

The Big Idea

When studying a function, we often want to answer three important questions:

Is the function increasing or decreasing?
Does the function have any local maxima or minima?
Where does the graph change its shape or curvature?

These questions can be answered using the first and second derivatives.

The first derivative helps us understand the slope of a graph, while the second derivative helps us understand its concavity.

In this post, we will use the function:

$f(x)=x^4-2x^3$

to understand these concepts.

Understanding the First Derivative

What Is the First Derivative?

The first derivative measures the rate at which a function changes.

It tells us whether the graph is moving upward or downward as we move from left to right.

The first derivative is written as:

$f'(x)=\frac{d}{dx}f(x)$

Key Interpretation

If:

$f'(x)>0$

the function is increasing.

If:

$f'(x)<0$

the function is decreasing.

If:

$f'(x)=0$

the graph has a horizontal tangent and may contain a maximum or minimum.

Example

Consider the function:

$f(x)=x^4-2x^3$

Differentiating gives:

$f'(x)=4x^3-6x^2$

Factoring:

$f'(x)=2x^2(2x-3)$

To find important points on the graph, we set the derivative equal to zero:

$2x^2(2x-3)=0$

This gives:

$x=0,\quad x=\frac{3}{2}$

These values are called critical points.

Understanding Critical Points

What Is a Critical Point?

A critical point is a point where:

$f'(x)=0$

or where the derivative does not exist.

Critical points are important because maxima and minima can only occur at these points.

Important Note

Not every critical point is a maximum or minimum.

To determine whether a critical point is actually an extremum, we must check how the derivative behaves around it.

Understanding Relative Extrema

What Are Relative Extrema?

Relative extrema are local maximum and minimum points.

A relative maximum occurs when the graph rises and then falls.

A relative minimum occurs when the graph falls and then rises.

The First Derivative Test

We determine extrema by checking whether the derivative changes sign.

Relative Maximum

If:

$+\rightarrow-$

the function changes from increasing to decreasing.

Therefore, a relative maximum occurs.

Relative Minimum

If:

$-\rightarrow+$

the function changes from decreasing to increasing.

Therefore, a relative minimum occurs.

Applying the Test

For:

$f'(x)=2x^2(2x-3)$

we examine the intervals around the critical points.

For values less than 0:

$f'(x)<0$

For values between 0 and:

$\frac{3}{2}$

the derivative remains negative.

For values greater than:

$\frac{3}{2}$

the derivative becomes positive.

This gives the sign pattern:

$-\rightarrow-\rightarrow+$

Notice that at:

$x=0$

the derivative does not change sign.

Therefore:

$x=0$

is not an extremum.

At:

$x=\frac{3}{2}$

the derivative changes from negative to positive.

Therefore:

$x=\frac{3}{2}$

is a relative minimum.

Takeaway

A critical point becomes a relative extremum only when the derivative changes sign.

For this function, there is exactly one relative extremum.

Understanding the Second Derivative