Dr. Math Azzam Platform | High School & University Mathematics

Lesson overview

A limit answers one question: where is the function heading? In this lesson, you will answer using two kinds of evidence — a table of nearby values and the graph itself. The value exactly at the target may agree with the limit, differ from it, or not exist at all.

Explanation

Learning objectives

Interpret $\lim_{x\to a}f(x)=L$ as nearby behavior.
Estimate a limit numerically using values approaching from both sides.
Estimate left-hand, right-hand, and two-sided limits from a graph.
Distinguish $f(a)$ from $\lim_{x\to a}f(x)$ .
Recognize when a two-sided limit does not exist and explain why.

Definition

Limit

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

means that the outputs

f(x)

get closer to

L

as inputs

x

get closer to

a

from both sides.

Look near the point — not only at the point

The word approaches is the heart of a limit. Imagine walking toward a doorway: you can tell where you are heading before you stand exactly on the threshold.

Interactive lab

The line y equals x plus 3 approaches the point 2 comma 5 from both sides.x = 0.920 · f(x) = 3.92

Graph view: 1×

Drag to trace the curveApproaching from the left

Worked example · Read the destination

\displaystyle \lim_{x\to2}(x+3)

Trace the line toward x = 2 from the left and from the right.

Predict before you reveal

Evaluate

\lim_{x\to2}(x+3)

.

2

3

5

DNE

Common mistake. A limit is not asking “what is

x

?” and it is not automatically asking for

f(a)

. It asks what the nearby outputs approach.

Your turn

What does

\lim_{x\to4}f(x)=7

describe?

f(4)

must equal

7

Nearby outputs approach

7

x

approaches

7

The graph crosses

(7,4)

Estimate a limit numerically

Build evidence on both sides of the target. Choose inputs progressively closer to $a$ , then watch whether the outputs settle toward one number.

x<a\quad\longrightarrow\quad a\quad\longleftarrow\quad x>a

Worked example · Table method

f(x)=\dfrac{x^2-4}{x-2},\qquad \text{estimate }\lim_{x\to2}f(x)

Use values just below and just above 2. Do not substitute x = 2.

Predict before you reveal

Which inputs best estimate a two-sided limit at

x=2

?

Only

x=2

1,0,-1

1.99,1.999,2.001,2.01

Only values above

2

Interactive lab

Build a numerical limit table

f(x)=\dfrac{x^2-4}{x-2}

. Calculate each output, then estimate the limit as x approaches 2. Keep at least three decimal places where needed. Do not substitute x = 2.

Compare inputs on both sides of 2. Complete the output column.
Side	x	f(x)	Feedback
Left	1.9	________	—
Left	1.99	________	—
Left	1.999	________	—
Right	2.001	________	—
Right	2.01	________	—
Right	2.1	________	—

My limit estimate

Key takeaway

Mr. Azzam’s table routine

STEP 1

Bracket

Use values below and above the target.

STEP 2

Move closer

Shrink the distance to the target.

STEP 3

Compare

Look for one shared output.

Common mistake. Using values from only one side is not enough to justify a two-sided limit.

Your turn

A table near

x=3

shows outputs

5.9,5.99,6.01,6.1

as inputs close in from both sides. What is the best estimate of

\lim_{x\to3}f(x)

?

3

6

9

DNE

Estimate a limit graphically

Place your eyes on the graph to the left and right of the target input, then trace both branches inward. The height they approach is the limit.

Interactive lab

The line approaches an open point at 2 comma 4, while a filled point at 2 comma 1 gives the function value.x = 0.920 · f(x) = 2.92

Graph view: 1×

Drag to trace the curveApproaching from the left

Worked example · Read a hole correctly

\displaystyle \lim_{x\to2}f(x)

Ignore the filled point for a moment. Follow the nearby curve.

Predict before you reveal

Which feature shows the approached value on the hole graph?

The filled point at

(2,1)

The open point at

(2,4)

The vertical axisThe graph title

Your turn

On the hole graph above, what is

\lim_{x\to2}f(x)

?

1

2

4

DNE

Left-hand and right-hand limits

x\to a^-\;\text{means from the left}\qquad x\to a^+\;\text{means from the right}

A two-sided limit exists only when the two one-sided limits agree.

\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L

Interactive lab

A piecewise step approaches 2 from the left and 4 from the right at x equals 1.x = -0.440 · f(x) = 2

Graph view: 1×

Drag to trace the curveApproaching from the left

Worked example · A jump separates the two sides

\displaystyle \lim_{x\to1}f(x)

Read the left and right branches separately before deciding.

Predict before you reveal

On the jump graph, find

\lim_{x\to1^+}f(x)

.

1

2

4

DNE

Common mistake. Never average unequal one-sided limits. If the two sides disagree, the two-sided limit is DNE.

Your turn

On the jump graph, what is

\lim_{x\to1^-}f(x)

?

1

2

4

DNE

Limit versus function value

Open point = approached location. Filled point = actual function value. These can be different.

Interactive lab

The line approaches an open point at 2 comma 4, while a filled point at 2 comma 1 gives the function value.x = 0.920 · f(x) = 2.92

Graph view: 1×

Drag to trace the curveApproaching from the left

Nearby behavior

\lim_{x\to2}f(x)=4

At the point

f(2)=1

Worked example · Read the value and the limit separately

f(2)\quad\text{and}\quad\lim_{x\to2}f(x)

Use the same hole graph, but answer two different questions.

Predict before you reveal

On the hole graph, find

f(2)

.

1

2

4

Undefined

Common mistake. Do not read the filled point as the limit. One isolated point does not control nearby behavior.

Your turn

On the hole graph, which statement is correct?

f(2)=4

and the limit is

1

f(2)=1

and the limit is

4

Both equal

4

Both are undefined

When does a limit not exist?

Before writing DNE, identify the reason. In this lesson there are three important patterns.

Explanation

1 · Jump

Left and right approach different finite values.

A piecewise step approaches 2 from the left and 4 from the right at x equals 1.x = -0.440 · f(x) = 2

Graph view: 1×

Drag to trace the curveApproaching from the left

Explanation

2 · Unbounded behavior

Outputs grow without bound near a vertical asymptote.

The reciprocal curve has a vertical asymptote at x equals 1, falling without bound from the left and rising without bound from the right.x = -0.800 · f(x) = -0.556

Graph view: 1×

Drag to trace the curveApproaching from the left

Work it out. $f(0.99)=-100$ and $f(1.01)=100$ . Moving closer to $1$ makes the magnitudes grow, with opposite signs.

Explanation

3 · Oscillation

Outputs keep changing and never settle to one destination.

The graph of sine of 1 over x oscillates faster near zero and never settles.x = -0.432 · f(x) = -0.736

Graph view: 1×

Drag to trace the curveApproaching from the left

Work it out. The outputs stay in $[-1,1]$ , but peaks and troughs keep occurring closer to $0$ . Bounded does not mean settled.

Key takeaway

Decision routine

1. Find the left-hand behavior.

2. Find the right-hand behavior.

3. If both approach the same finite number, that is the limit.

4. Otherwise state DNE for a finite real limit and name the reason.

This routine checks for a finite real limit. Infinite-limit notation describes unbounded behavior, not a finite output. For this reciprocal, the two sides have opposite signs, so there is no single two-sided destination.

Your turn

Question 1 of 3

If

\lim_{x\to a^-}f(x)=2

and

\lim_{x\to a^+}f(x)=5

, what is

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

?

2

3.5

5

DNE

Limit Explorer · investigate it yourself

Now control the approach yourself. Move toward the target from both sides, compare numerical and graphical evidence, and make your prediction before checking.

Interactive lab

Explore the Limit

f(x)=(x^2-4)/(x-2),\quad x\ne2

. Move x toward 2 from both sides, compare the outputs, then predict the limit. The hole is not a function value.

Horizontal axis: x from 0 to 4. Vertical axis: f(x) from 0 to 6 (independent scales).

Approach from left

x\to2

Approach from right

Current x

1.50

f(x)

3.50

|x − 2|

0.50

Side

Left (−)

As x approaches 2, which value does f(x) approach?

After comparing both sides in the explorer, use the separate Practice area to apply these ideas independently.

Continue to Practice

Lesson overview

Look near the point — not only at the point

Worked example · Read the destination

1. Approach from the left

2. Approach from the right

3. Compare

4. Conclusion

Your turn

Your turn · student questions

Estimate a limit numerically

Worked example · Table method

1. Choose nearby inputs

2. Read the outputs

3. Left-hand trend

4. Right-hand trend

5. Estimate

Your turn

Your turn · student questions

Estimate a limit graphically

Worked example · Read a hole correctly

1. Trace from the left

2. Trace from the right

3. Compare

4. Two-sided limit

Your turn

Your turn · student questions

Left-hand and right-hand limits

Worked example · A jump separates the two sides

1. Left-hand limit

2. Right-hand limit

3. Compare

4. Conclusion

Your turn

Your turn · student questions

Limit versus function value

Worked example · Read the value and the limit separately

1. Read the filled point

2. Trace both nearby branches

3. Report the answers separately

Your turn

Your turn · student questions

When does a limit not exist?

1 · Jump

2 · Unbounded behavior

3 · Oscillation

Your turn

Your turn · student questions

Limit Explorer · investigate it yourself