Numerical, Graphical, and Symbolic Limits

The Central Idea

Suppose \(f(x)\) gets closer and closer to a number \(L\) as \(x\) gets closer and closer to a number \(a\). We write

Read this as: the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\).

The statement is about what happens near \(x=a\), not necessarily what happens exactly at \(x=a\).

Limits Describe Approach

The arrow in

means that \(x\) is allowed to move closer and closer to \(a\).

At the same time, we observe what happens to

If the output values settle toward one number, that number is the limit.

A First Example

Consider the function

As \(x\) approaches \(3\),

approaches

Therefore,

Why Limits Matter

Limits provide the basic language for describing quantities that change continuously or nearly continuously.

They allow us to study:

• instantaneous rates of change;
• slopes of tangent lines;
• areas accumulated from very small pieces;
• continuity and discontinuity;
• asymptotic behavior;
• functions near points where direct substitution fails.

Derivatives and integrals are built from limits.

Three Ways to Study a Limit

We will use three main representations.

1. Numerical: examine values in a table.
2. Graphical: inspect the graph near the target input.
3. Symbolic: use algebraic expressions and known rules.

These approaches should reinforce one another rather than compete with one another.

Numerical Limits

A numerical approach samples function values at inputs increasingly close to the target input.

For example, suppose

and we want

We can inspect values near \(x=2\).

Approaching from Below

Choose inputs slightly less than \(2\):

The outputs are moving toward \(4\).

Approaching from Above

Now choose inputs slightly greater than \(2\):

These outputs also move toward \(4\).

So the numerical evidence supports

Use Values on Both Sides

When estimating a two-sided limit numerically, inspect values on both sides of the target input.

For a target \(a\), useful inputs might look like

and

The key question is whether the corresponding output values approach the same number.

The Function Value Is Not the Same Question

Suppose a function satisfies

but values near \(x=2\) approach \(4\).

Then

even though

A limit and a function value are related concepts, but they are not automatically identical.

A Limit Can Exist When the Function Is Undefined

Consider

Direct substitution at \(x=2\) gives

so the original expression is undefined there.

But for \(x\neq2\),

Therefore, as \(x\) approaches \(2\),

Thus,

Numerical Table for a Removable Hole

The same limit can be estimated numerically.

The values clearly approach \(4\) from both sides.

Graphical Limits

A graph lets us see nearby behavior directly.

To estimate

trace the graph toward \(x=a\) from the left and from the right.

If the \(y\)-values approach the same height \(L\) from both sides, then

Reading a Graph Near a Hole

A graph may contain an open circle at

The open circle indicates that the graph does not include that point.

But if the curve approaches the open circle from both sides, then

The missing point does not destroy the limit.

A Filled Point Can Be Somewhere Else

Imagine a graph with:

• an open circle at \((2,4)\), and
• a filled point at \((2,7)\).

Then

but if the surrounding curve approaches \(4\),

Again, the function value and the limit answer different questions.

Graphical Limit Checklist

When reading a limit from a graph:

1. Locate the target input \(x=a\).
2. Follow the graph toward \(a\) from the left.
3. Follow the graph toward \(a\) from the right.
4. Identify the \(y\)-value approached from each side.
5. Ignore the actual plotted value at \(x=a\) until the limit is settled.
6. If both sides approach the same value, that value is the limit.

When a Graph Shows No Two-Sided Limit

Suppose the graph approaches \(2\) from the left and \(5\) from the right.

Then there is no single number approached from both directions.

Therefore,

Later we will describe the two directions separately using one-sided limits.

Symbolic Limits

A symbolic approach works directly with the function formula.

For many familiar functions, direct substitution gives the limit immediately.

For example,

Direct Substitution

If the function behaves regularly at the target input, try substituting the target value.

For example,

becomes

Therefore,

Direct Substitution Is a First Check

Direct substitution is often the fastest first step, but it does not solve every limit.

If substitution gives an ordinary real number, that value is often the limit for the elementary functions studied at this stage.

If substitution produces an undefined or ambiguous expression such as

more work is needed.

Why \(\frac{0}{0}\) Is Not the Answer

The expression

is not a numerical value.

In a limit problem, it often signals that the original expression can be rewritten so that nearby behavior becomes clearer.

For example,

gives \(0/0\) by direct substitution, but factoring reveals the actual limit \(4\).

Symbolic Simplification Can Reveal Nearby Behavior

Consider

Factor the numerator:

For \(x\neq3\),

Therefore,

Canceling Does Not Redefine the Original Function

In

canceling \(x-3\) is valid only when

That is enough for the limit, because a limit concerns values arbitrarily close to \(3\), not necessarily the value at \(3\) itself.

Numerical, Graphical, and Symbolic Agreement

A well-understood limit often tells the same story in all three representations.

For

• the numerical values approach \(6\);
• the graph approaches the point \((3,6)\) but has a hole there;
• the algebra simplifies to \(x+3\) for \(x\neq3\).

All three approaches support

Why Use More Than One Representation?

Different representations expose different features.

• Tables are useful for estimation and numerical evidence.
• Graphs are useful for visualizing approach and discontinuity.
• Symbolic expressions can provide exact values and explanations.

When one representation is unclear, another may resolve the ambiguity.

Limits and Nearby Inputs

The word nearby is central.

To study

we might examine inputs such as

We are not limited to those particular numbers. Any sequence of inputs moving toward \(5\) can provide evidence about the same local behavior.

Getting Closer Matters More Than Taking Many Random Samples

A table is most useful when the inputs actually approach the target.

Values such as

do not provide especially strong evidence about a limit at \(x=5\).

Values such as

are more informative because they reveal what the function is doing increasingly close to the target.

Do Not Confuse Approach with Equality

If

we are not saying

We are saying that \(x\) can be taken as close to \(2\) as needed.

Likewise, if

the outputs are approaching \(7\). They do not need to equal \(7\) at every nearby input.

A Limit Is Local Information

A limit near \(x=a\) depends mainly on what the function does near \(a\).

The function can behave very differently far away without changing the limit.

This is why limits are a form of local analysis.

Example: A Piecewise Function

Suppose

Near \(x=2\), the function follows

Therefore,

even though

Example: Matching Table and Formula

Consider

For \(x\neq1\),

A table near \(x=1\) gives:

Numerically and symbolically,

Estimating from Rounded Data

Numerical data may be rounded, so a table does not always reveal the exact limit immediately.

Suppose outputs near \(x=4\) are

from one side and

from the other.

The natural estimate is

Numerical Evidence Is Evidence, Not Always Proof

A table can strongly suggest a limit, but finitely many sampled values do not by themselves guarantee what happens at every nearby input.

Symbolic reasoning or a formal limit argument can provide stronger justification.

Numerical estimation remains valuable for developing intuition, checking work, and investigating functions that are difficult to simplify.

Graphical Evidence Also Has Limits

A graph depends on scale, resolution, and the window being viewed.

Two curves can look nearly identical at one scale but behave differently when magnified.

Therefore, use a graph as an analytical tool, but do not assume that a picture alone proves every exact limit.

Symbolic Reasoning Can Also Require Care

Algebraic simplification must respect domain restrictions.

If

is simplified to

the simplification is valid only for

That distinction is often exactly what produces a removable discontinuity.

A Practical Workflow

1. Identify the target input \(a\).
2. Try direct substitution.
3. If substitution gives a normal value, record the likely limit.
4. If substitution fails, inspect nearby numerical values.
5. Use the graph if one is available.
6. Look for algebraic simplification.
7. Compare the three representations.
8. State the limit only when the evidence supports one common approached value.

Example: Full Three-Representation Workflow

Evaluate

Direct substitution gives

So direct substitution alone does not finish the problem.

Factor:

For \(x\neq4\),

Therefore,

A table would show outputs approaching \(8\), and the graph would look like the line \(y=x+4\) with a hole at \((4,8)\).

Common Error: Plugging In and Stopping at \(\frac{0}{0}\)

If substitution gives

do not report \(0\), do not report \(1\), and do not treat \(0/0\) as the final result.

Instead, interpret it as a signal to investigate further.

Common Error: Using Only One Side of a Table

A two-sided limit requires compatible behavior from both sides.

Values approaching \(a\) only from below may not reveal what happens from above.

Always inspect both directions unless the problem specifically asks for a one-sided limit.

Common Error: Reporting the Filled Dot Instead of the Limit

On a graph, the filled point at \(x=a\) tells you

It does not automatically tell you

Trace the nearby graph before reading the function value.

Common Error: Thinking the Function Must Reach the Limit

A function can approach a value without taking that value at the target input.

A removable hole is the standard example.

Limits describe approach, not mandatory arrival.

Common Error: Cancelling an Additive Term

From

we may factor

and then cancel the factor \(x-2\).

We may not simply cancel the \(x\) or the \(2\) across addition and subtraction.

Cancellation applies to common multiplicative factors.

Common Error: Assuming Every Limit Exists

Some functions approach different values from different sides.

Others become arbitrarily large in magnitude.

Still others oscillate without settling toward one value.

The later lessons in this section will classify these behaviors more carefully.

Practice 1: Direct Substitution

Evaluate:

Practice 2: Polynomial Limit

Evaluate:

Practice 3: Numerical Interpretation

A table gives the following values near \(x=5\):

Estimate

Practice 4: Factoring a Difference of Squares

Evaluate:

Practice 5: Function Value versus Limit

Suppose a graph approaches \(y=6\) from both sides as \(x\to2\), but the graph has a filled point at \((2,9)\).

Find:

and

Practice 6: Hole in a Rational Function

Evaluate:

Practice 7: Does the Two-Sided Limit Exist?

A graph approaches \(3\) as \(x\) approaches \(4\) from the left and approaches \(7\) as \(x\) approaches \(4\) from the right.

Does

exist?

Practice 8: Interpret the Notation

Explain in words what

means.

Practice 9: Identify the Misconception

A student says: “If \(f(3)\) is undefined, then \(\lim_{x\to3}f(x)\) must also be undefined.”

Explain why this statement is not generally correct.

Practice 10: Full Workflow

Evaluate:

Describe how a numerical table and graph would support the symbolic result.

Practice Answers

Practice 1

Practice 2

Practice 3

The values from both sides approach \(12\), so

Practice 4

For \(x\neq5\),

Therefore,

Practice 5

The surrounding graph approaches \(6\), so

The filled point gives

Practice 6

Thus, for \(x\neq1\),

Therefore,

Practice 7

The two sides approach different values, \(3\) and \(7\).

Therefore,

Practice 8

As \(x\) gets closer and closer to \(7\), the values of \(g(x)\) get closer and closer to \(12\).

Practice 9

A limit depends on values near the target input, not necessarily on the function value at the target itself. A function can have a hole at \(x=3\) while still approaching one well-defined number from both sides.

Practice 10

For \(x\neq6\),

Therefore,

A numerical table would show outputs approaching \(12\) from both sides. The graph would look like the line \(y=x+6\) with a hole at \((6,12)\).

What to Remember

A limit describes the value a function approaches:

• Numerically, inspect values increasingly close to the target input.
• Graphically, trace the curve from both sides.
• Symbolically, begin with direct substitution and simplify when necessary.
• The value \(f(a)\) does not automatically determine the limit.
• A limit can exist even when \(f(a)\) is undefined.
• A two-sided limit requires compatible behavior from both directions.
• Tables and graphs provide strong evidence; symbolic reasoning can often give an exact result.

The central habit is to ask: what value are the outputs approaching as the inputs approach the target?

Where We Go Next

This lesson treated limits from both sides as a single idea. Sometimes the two directions behave differently, and sometimes function values grow without bound.

The next lesson, One-Sided and Infinite Limits, separates left-hand and right-hand behavior and introduces limits associated with unbounded growth.