Limits from Graphs | AP Calculus Topic 1.3 Study Notes

1. Read the approach, not just the point

Estimating a limit from a graph means reading the output height the curve approaches as the input moves toward a target. The input lies on the horizontal axis; the limit is an output read from the vertical axis.

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

For a finite two-sided limit, the curve must approach the same finite height from both sides. The actual value $f (a)$ may agree, differ, or be undefined. A filled point answers the function-value question; nearby branches answer the limit question.

Quick rule: follow the curve, not the dot.

Read the left-hand approach, read the right-hand approach, then compare. Do not substitute the height of an isolated filled point for the limiting height.

AP® Calculus AB and BC Topic 1.3 emphasizes one-sided limits, graphical estimates, reasons limits fail to exist, and the limitations of a graph’s scale. If the notation is new, review Topic 1.2: defining limits and using limit notation.

2. Interactive graph limit trainer

Choose one of the five scenarios. Read the left-hand limit, right-hand limit, two-sided limit and function value before revealing the explanation.

Choose graph type

Target: $x \to 2$

Line approaching an open point at (2,4) from both sides, with a filled point at (2,1). — Read the axis values and distinguish open and filled points. Horizontal and vertical scales differ.

View the function model

f (x) = x + 2, x \neq 2

f (2) = 1

3. One-sided limits from graphs

Left-hand limit

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

Trace the branch at inputs $x < a$ toward the target. Inputs such as $a - .1$ , $a - .01$ and $a - .001$ are progressively closer from the left.

Right-hand limit

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

Trace the branch at inputs $x > a$ toward the target. Inputs such as $a + .1$ , $a + .01$ and $a + .001$ approach from the right.

The superscript tells you the side of the input target, not the sign of the output or whether the inputs are positive.

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

When both sides approach the same finite value, $lim_{x \to a} f (x) = L$ . Distinct finite one-sided limits give a jump and no two-sided limit. At a domain endpoint, only the available side may be investigated; label that limit as one-sided.

4. Limit versus function value

The limit describes the graph’s nearby behavior. The function value describes the actual output assigned at the target. These are different questions, even when the answers happen to agree.

Separate the two readings
Quantity	What to inspect	What it answers
$lim_{x \to a} f (x)$	Trace the branches near the target.	What height is approached?
$f (a)$	Find the filled point at the target input.	What output is actually defined?

For example, suppose the curve approaches a height of three from both sides near input two, but the filled point is $(2, 5)$ . Then $lim_{x \to 2} f (x) = 3$ , while $f (2) = 5$ .

An open point excludes that location on a branch. It does not automatically make the limit fail, and it does not guarantee that the function is undefined at that input: a filled point may exist at another height.

5. Graph symbols and what they mean

Symbols used in graphical limit questions
Feature	Meaning	Effect on the reading
Open circle	The displayed point is excluded from that branch.	A finite limit can still exist when nearby branches approach the same height.
Filled circle	The function includes this point.	It gives the function value, not automatically the limit.
Jump	Distinct finite one-sided limiting heights.	No two-sided limit exists.
Vertical asymptote	At least one side becomes unbounded near an input.	Inspect each side’s direction; a common finite limit does not exist.
Horizontal asymptote	A horizontal line approached at one or both ends.	Describes end behavior; inspect each end separately.
Persistent oscillation	The curve does not settle near one height.	No finite limit exists if the non-settling behavior continues arbitrarily close to the target.

A dashed vertical line is a guide, not a branch of the function. A curve leaving the viewing window is a clue, not proof by itself that values grow without bound.

6. Five common graph scenarios

A hole or removable discontinuity

Both sides approach a common finite height, but the point is missing or assigned a different value. The limit exists. Changing just the target’s value does not change the limit.

A jump

The left and right branches approach different finite heights. Each one-sided limit can exist even though the two-sided limit does not.

A vertical asymptote

At least one side increases or decreases without bound. If both sides increase without bound, use positive-infinity notation. If both decrease without bound, use negative-infinity notation. Opposite infinite directions do not produce a single two-sided infinite limit.

Oscillation

The function $\sin (\frac{1}{x})$ , with angles in radians, continues oscillating near zero. Peaks and troughs persist arbitrarily close to the target, so no limiting height is approached. A finite sketch cannot draw infinitely many waves; inspect the stated behavior, not just the last visible wiggle.

Wiggling alone is not enough to rule out a limit. If the oscillations shrink toward one height, the limit may exist; for example, $lim_{x \to 0} x \sin (\frac{1}{x}) = 0$ .

A continuous graph

If a graph is continuous at the target, the left-hand limit, right-hand limit, two-sided limit and function value agree. The graph need not be smooth: a continuous corner can still have a limit. A corner concerns the slope, a different question.

7. A step-by-step graph-reading method

Locate the target input. Find $x = a$ on the horizontal axis and check the axis increments.
Trace from the left. Follow the branch with $x < a$ toward the target. Record the approached output height.
Trace from the right. Follow the branch with $x > a$ toward the target. Record its approached height.
Compare the approaches. A finite two-sided limit requires the same finite height from both sides.
Read the function value separately. Look for the filled point only when answering the function-value question.
Check the viewing window. Look for hidden holes, compressed scales, off-screen branches and oscillation near the target.

Use a complete conclusion.

State both one-sided limits and whether they agree. If the two-sided limit fails, identify the reason rather than writing only “DNE.” Use “approximately” when the plotted scale supports an estimate rather than an exact labelled value.

8. Scale can hide important behavior

A graph that looks continuous when zoomed out may contain a small hole or jump. A compressed vertical scale can make a vertical asymptote look like an ordinary steep curve. Limited resolution can conceal repeated oscillations.

Check the value represented by each tick mark on both axes.
Look for open and filled points at the same input.
Inspect continuation arrows and asymptote guides.
Zoom locally on a digital graph, and compare several viewing windows.
Read the caption, domain and any definition supplied with the graph.
Use a table or algebra when available, while remembering that finite samples can also miss behavior.

A graph is evidence at a chosen scale. A steep line is not automatically a vertical asymptote; leaving the top of a window does not alone establish infinite behavior. Likewise, a finite number of waves does not by itself establish that oscillation persists arbitrarily close to an input.

When the plotted resolution does not determine a unique value, report a sensible graphical estimate instead of inventing extra decimal places. Continue with Topic 1.4: estimating limits from tables to connect visual and numerical evidence.

9. Worked examples with graph readings

Example 1: a hole with a different function value

The branches approach $(2, 4)$ from both sides, but the filled point is $(2, 1)$ .

lim_{x \to 2^{-}} f (x) = 4, lim_{x \to 2^{+}} f (x) = 4

lim_{x \to 2} f (x) = 4, f (2) = 1

This is a removable discontinuity. The limit exists because nearby outputs from both sides approach four. The filled point does not pull the limit down to one.

Example 2: a jump discontinuity

The left branch approaches three near input one, while the right branch approaches negative two.

Left branch approaches the open point (1,3); right branch begins at the filled point (1,-2). — The left limit is three and the right limit is negative two; the filled right endpoint also assigns the actual value.

lim_{x \to 1^{-}} f (x) = 3, lim_{x \to 1^{+}} f (x) = −2

Since $3 \neq −2$ , the two-sided limit does not exist. For this displayed model, $f (1) = −2$ . The value at the target cannot make the two approaches agree.

Example 3: a vertical asymptote

The curve increases without bound on both sides of zero.

Two separate branches of one over x squared increase without bound on both sides of the vertical line at zero. — Both branches continue upward near zero. The vertical dashed line is an asymptote, not part of the graph.

lim_{x \to 0^{-}} f (x) = \infty, lim_{x \to 0^{+}} f (x) = \infty

The two-sided infinite behavior is positive infinity, but there is no finite real-number limit. For the displayed model $f (x) = \frac{1}{x^{2}}$ , the function is undefined at zero. A function could be assigned a separate value there without changing its infinite nearby behavior.

Example 4: a continuous function

The displayed line is continuous through $(5, 7)$ .

Continuous line through the filled point (5,7). — At input five, both approached heights and the actual function value are seven.

lim_{x \to 5^{-}} f (x) = lim_{x \to 5^{+}} f (x) = lim_{x \to 5} f (x) = f (5) = 7

All four readings agree. Continuity justifies using the point’s height for the limit in this case; merely seeing a filled dot would not be enough.

10. Limits at infinity from graphs

This retained extension looks at end behavior rather than approach to a finite input. For $lim_{x \to \infty} f (x)$ , inspect the far-right trend; for $lim_{x \to - \infty} f (x)$ , inspect the far-left trend.

If the curve approaches a horizontal line $y = L$ at one end, that end’s limit is the height of the line. The two ends can differ. A horizontal asymptote can be crossed; it is not a boundary the curve must stay on one side of.

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

If the curve increases without bound at an end, use positive-infinity notation when justified; if it decreases without bound, use negative-infinity notation. Oscillation that never settles can prevent an end-behavior limit too. An ordinary finite viewing window may only suggest an end trend, so use the stated function or annotations for conclusions about arbitrarily large inputs.

11. Three main reasons no finite limit exists

How to justify DNE
Reason	What the graph shows	Conclusion
Different finite one-sided limits	The two branches approach distinct heights.	The two-sided limit does not exist; this is a jump.
Unbounded behavior	One or both sides increase or decrease without bound.	No finite real limit exists. State the appropriate infinite behavior, if established.
Persistent oscillation	Outputs do not settle near a single height arbitrarily close to the target.	No limiting height exists on the affected side; a finite two-sided limit fails.

A hole is not automatically DNE.

If both sides approach one finite height, the limit exists despite an excluded point or a different assigned value. Distinguish this from a jump, unbounded behavior or persistent oscillation.

12. AP Calculus tips and quick reference

Check both sides for a two-sided finite limit.
Read output heights rather than answering with the target input.
Keep the function value separate from nearby approach behavior.
Use correct notation with the left or right superscript where needed.
Explain DNE by naming the disagreement, unbounded behavior or persistent oscillation.
Respect scale and the uncertainty of an approximate reading.
Read arrows and dashed lines carefully: they are graphical clues whose meaning depends on the surrounding function and labels.

Graphical limit reference
Notation	How to read it	Meaning
$lim_{x \to a^{-}} f (x)$	Trace toward the target using smaller inputs.	The output height approached from the left.
$lim_{x \to a^{+}} f (x)$	Trace toward the target using larger inputs.	The output height approached from the right.
$lim_{x \to a} f (x)$	Compare both sides.	A common finite height, if one exists.
$f (a)$	Find the filled point at the target.	The actual function value.
$lim_{x \to \infty} f (x)$	Inspect the far-right trend.	End behavior as the input increases without bound.
$lim_{x \to - \infty} f (x)$	Inspect the far-left trend.	End behavior as the input decreases without bound.

For practice, write four separate readings for each graph: left-hand limit, right-hand limit, two-sided limit and function value. Then explain why the readings agree or differ.

13. Graphical limit FAQs

How do you find a limit from a graph?

Locate the target input, trace the nearby branches from each side, and compare the approached output heights. A finite two-sided limit requires the same finite height from both sides.

Does an open circle mean the limit does not exist?

No. It excludes that point on the branch. The limit can still exist if nearby branches approach a common finite height; a separate filled point may give the actual value at that input.

What is the difference between a limit and a function value?

The limit $lim_{x \to a} f (x)$ describes nearby approach behavior. The value $f (a)$ is the output assigned at the target. One can exist without the other, or the values can differ.

When does a two-sided limit not exist from a graph?

Common reasons are unequal finite one-sided limits, unbounded behavior or oscillation that fails to settle arbitrarily close to the target. State the relevant reason.

How do you read a left-hand limit from a graph?

Follow the branch at inputs $x < a$ toward the target. Its approached height is $lim_{x \to a^{-}} f (x)$ . The word left refers to input values, not negative outputs.

How do you read a right-hand limit from a graph?

Follow the branch at inputs $x > a$ toward the target. Its approached height is $lim_{x \to a^{+}} f (x)$ .

14. Continue learning

Use notation, graphs and tables together. Topic 1.2 explains how to read limit statements; Topic 1.4 develops numerical estimates. Later Unit 1 topics study algebraic methods, infinite limits, limits at infinity and continuity in more detail.

The test collection covers the full Limits and Continuity unit. Study the later topics before using it as a complete unit assessment.

0000003998 : Abdad Alam Shamim Alam

This is the bestest institution I have ever come to and I love it very much.

Aliki S

Ali

Sly is him

Jitendra Kumar Kumawat

Raahil Hasan

Tina Mudarres

The classes are amazing and my child learnt so much

alisha gadoya

I have experienced a lot of good things, it has taught me so many things and from B/Cs I have gone to an A! This is wonderful and Mavish’s class is awesome.

smasher 123

It’s sensational my kid went. First he only would get C now it’s all A’s really good and recommend

Mosa Al- Samaraie

AMAZING PRICES GREAT TEACHERS STRAIGHT FORWARD LEARNING STEADY PACE IN TUTORING

Cael Dagnelie

I had a great experience learning math with Ms. Mavish. She explains complex topics in a very clear and simple way, which made it easier for me to understand and enjoy the subject. Her patience and dedication really stood out, and she always made sure that everyone in the class was keeping up. I especially appreciated how approachable she was — I never felt afraid to ask questions, and she was always willing to help. Thanks to her teaching, my confidence in math has grown a lot. I’m really thankful for the effort she puts into every lesson!