Introduces the limit as what a function's outputs approach near a point, distinct from its value there. Estimates limits from tables and graphs, distinguishes one-sided from two-sided limits, and identifies the situations in which a limit fails to exist.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 12 — Introduction to Calculus
§12.1 Finding Limits: Numerical and Graphical Approaches, pp. 1388-1403
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1388-1403 — the pages these objectives are drawn from
Warm-up
A function can be undefined at a point and perfectly well behaved around it.
Discussion prompt
A function equals the input plus one everywhere except at two, where it is undefined. What happens near two?
Hint: What do the outputs do as the inputs close in?
Answer:
At inputs of 1.9 and 2.1 the outputs are 2.9 and 3.1. At 1.99 and 2.01 they are 2.99 and 3.01.
So the outputs close in on 3 from both sides, even though the function has no value at 2 at all.
That number is the limit. It describes what the function approaches rather than what it equals, and the distinction is what the whole chapter rests on.
Concept
A limit describes the value a function's outputs close in on as the inputs approach a point, deliberately ignoring what happens at the point itself.
limit — the value a function's outputs approach as its inputs approach a given point, regardless of the value at that point
\[ \lim_{x\to 2}f(x)=3 \]
Ignoring the point itself is not a limitation but the whole design. It is what allows a limit to describe behaviour at a hole, at a jump, or anywhere the function is undefined.
Figure (svg): A graph with a hole at one point, showing the curve approaching a value the function does not attain there
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1388-1392
Section
Section 1
Concept
A function's value at a point and its limit there are different questions, and either can exist without the other.
The two agreeing is the ordinary case and has its own name — continuity, which §12.3 develops. The interesting limits are precisely the ones where they differ or where the value is missing.
Figure (svg): A contrast between the value of a function at a point and the limit of the function as it approaches that point
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1388-1393
Picture it
Two different questions about the same point.
Figure (svg): A contrast between the value of a function at a point and the limit of the function as it approaches that point
The last row on each side is the design decision. Ignoring the point is what makes the limit useful, since the interesting cases are exactly where the point misbehaves.
Worked example
The value is missing and the limit is not.
\[ \text{A function equals } x+1 \text{ except at } x=2, \text{ where it is undefined. Find the limit there.} \]
Check the value
Why: Undefined at two.
Look at nearby inputs
Why: From both sides.
\[ \text{close to } 3 \]
Note the limit ignores the point
Why: Only nearby matters.
State the limit
Why: What the outputs approach.
\[ 3 \]
Figure (svg): A graph with a hole at one point, showing the curve approaching a value the function does not attain there
\[ \lim_{x\to 2}f(x)=3 \]
Verify: check both sides agree
Why: From below the outputs approach 3 and from above they do too, so the two-sided limit exists. The missing value at 2 played no part in the reasoning, which is exactly what the definition intends.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1389-1391
Prediction
A function's value at a point is changed to something else.
Predict first
What happens to the limit there?
Correct: Nothing; the limit ignores the point.
Why: A limit is determined entirely by nearby inputs, so the value at the point plays no part. That independence is what allows a limit to exist where the function is undefined.
Worked example
Both exist and they differ.
\[ \text{A function equals } x+1 \text{ except at } x=2, \text{ where it equals } 7. \text{ Compare the value and the limit.} \]
Find the value
Why: Stated directly.
\[ 7 \]
Find the limit
Why: From nearby inputs.
\[ 3 \]
Compare
Why: They differ.
Note which is which
Why: The limit ignores the point.
\[ \lim\text{ is } 3 \]
Figure (svg): The solution to Worked example value and limit disagreeing shown as a ladder of expressions, one row per legal move
\[ f(2)=7, \; \lim_{x\to 2}f(x)=3 \]
Verify: check the limit is unaffected
Why: Changing the value at 2 to any number at all leaves the limit at 3, since the limit never looks at that point. That independence is the clearest demonstration that the two are genuinely different questions.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1391-1393
Trap
\[ \lim_{x\to 2}f(x)=f(2) \]
Substitute the point into the function
Why: The limit is treated as the value.
Where the function is undefined there is nothing to substitute, and where the two disagree the answer is wrong.
The limit is about nearby inputs, not the point itself. Substitution answers a different question.
It happens to give the right answer when the two agree, which is the ordinary case — but that agreement is a property to check rather than assume.
The interesting limits are exactly the ones where substitution fails, which is why the technique exists at all.
Sorting
Value or limit.
Sort into buckets
Sort each description.
Faded example
A function equalling x plus one except at two.
Fill in the blanks
\lim_3f(x)=undefined, \text___f(2)\text______
Why: The limit comes from nearby inputs, where the function does equal x plus one. The missing value at the point itself is irrelevant to the limit by definition.
Explain it to yourself
It seems strange to exclude the point being approached.
Discussion prompt
Explain why the definition is set up that way.
Hint: What would including it prevent?
Answer:
Including the point would make the limit undefined wherever the function is — which is precisely the case the idea was invented for.
A hole, a jump, or a removed value are all situations where the surrounding behaviour is perfectly clear and the point itself says nothing useful.
So excluding it is what makes limits able to describe those cases. A good explanation notes that when the two do agree, that agreement is itself a property worth naming — which is continuity.
Section
Section 2
Concept
Choosing inputs progressively closer to the point, from below and from above, shows what the outputs are converging on.
The table's limitation is real: it shows a pattern rather than establishing one, and a function can behave unexpectedly closer in than any table reaches. That is why the algebraic techniques of §12.2 matter.
Figure (svg): A table of inputs approaching a value from both sides, with the corresponding outputs converging on a common number
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1393-1397
Picture it
Two columns converging on one value.
Figure (svg): A table of inputs approaching a value from both sides, with the corresponding outputs converging on a common number
The convergence from both sides is what the table is demonstrating. One side alone would establish only a one-sided limit.
Worked example
Approach from both directions.
\[ \text{Estimate } \lim_{x\to 2}\frac{x^2-4}{x-2} \text{ from a table.} \]
Approach from below
Why: At 1.9, 1.99, 1.999.
\[ 3.9, 3.99, 3.999 \]
Approach from above
Why: At 2.1, 2.01, 2.001.
\[ 4.1, 4.01, 4.001 \]
Compare the two
Why: Both converge.
\[ \text{on } 4 \]
State the estimate
Why: The common value.
\[ 4 \]
Figure (svg): A table of inputs approaching a value from both sides, with the corresponding outputs converging on a common number
\[ \lim_{x\to 2}=4 \]
Verify: check algebraically
Why: The numerator factors as x minus two times x plus two, and cancelling leaves x plus two — which is 4 at x equal to 2. The table's estimate is confirmed exactly, which is what §12.2's techniques will do routinely.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1394-1396
Prediction
Both columns converge on the same value.
Predict first
What does that show?
Correct: Strong evidence for the limit, but not proof.
Why: A table samples finitely many inputs while a limit concerns all inputs arbitrarily close. The evidence is convincing when the pattern is clear, but a function can behave unexpectedly closer in than the table reaches.
Worked example
Evidence is not proof.
\[ \text{Why is a table only evidence for a limit?} \]
Note what a table shows
Why: Finitely many inputs.
Note what a limit claims
Why: Behaviour arbitrarily close.
Compare
Why: The sample cannot cover them all.
Conclude
Why: A pattern, not a proof.
Figure (svg): The solution to Worked example what a table cannot settle shown as a ladder of expressions, one row per legal move
\[ \text{suggestive but not conclusive} \]
Verify: consider a function that would fool a table
Why: A function oscillating rapidly very close to the point can look convergent at every table value while having no limit at all. That is why the algebraic methods matter, and why a table is a first step rather than a final answer.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1396-1397
Error analysis
A student estimates a limit from a table.
Annotate
On: \( x=1.9,\;1.99,\;1.999 \;\Longrightarrow\; \text{the limit is }3 \)
Sampling one side establishes a one-sided limit, which is a weaker statement. A jump discontinuity looks perfectly convergent from either side taken alone.
Faded example
Outputs at inputs approaching from below.
Fill in the blanks
2.9,\;2.99,\;2.999 \;\to\; 3
Why: Each additional digit brings the output closer to 3, which is the value being approached. The same pattern from above would confirm the two-sided limit.
Sorting
Both sides are needed.
Sort into buckets
Sort each situation.
Step zero
You are estimating a limit numerically.
Discussion prompt
What do you choose before computing anything?
Hint: Inputs from where?
Answer:
Inputs from both sides, progressively closer to the point. One side alone establishes only a one-sided limit.
Each row should be closer than the last, so the convergence is visible as a pattern rather than a single value.
And remember what the table can and cannot show: it is evidence for a limit, not a proof. A good habit is to treat the estimate as a prediction that the algebraic methods of §12.2 will confirm.
Section
Section 3
Concept
Following the graph towards a point from each side shows what height the curve is heading for, which is the limit from that side.
The open circle convention matters: it marks a point the graph approaches but does not include. A filled dot elsewhere on the same vertical line shows where the function's value actually is, and the two can be at different heights.
Figure (svg): A graph with a hole at one point, showing the curve approaching a value the function does not attain there
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1397-1400
Picture it
The open circle is the missing value.
Figure (svg): A graph with a hole at one point, showing the curve approaching a value the function does not attain there
The curve heads for that height from both sides, so the limit is that height — regardless of the circle being open. The graph shows both facts at once.
Worked example
The graph shows both.
\[ \text{A graph approaches height } 3 \text{ at } x=2 \text{ with an open circle there, and a filled dot at height } 5. \text{ Read both.} \]
Trace from the left
Why: Heading for three.
\[ 3 \]
Trace from the right
Why: Also three.
\[ 3 \]
State the limit
Why: Both agree.
\[ 3 \]
Read the value
Why: The filled dot.
\[ 5 \]
Figure (svg): A graph with a hole at one point, showing the curve approaching a value the function does not attain there
\[ \lim=3, \; f(2)=5 \]
Verify: check the two conventions
Why: The open circle marks a height the curve approaches but does not attain, and the filled dot marks where the function actually is. Both appear on the same vertical line, which is how a graph shows a limit and a value disagreeing.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1398-1399
Prediction
A graph has an open circle at the point being approached.
Predict first
Does the limit exist?
Correct: Yes, if both sides head for that height.
Why: The limit ignores the point itself, so a missing value is irrelevant. What matters is whether the curve approaches the same height from both directions.
Worked example
The two sides head for different heights.
\[ \text{A graph rises to height } 1 \text{ from the left and sits at height } 3 \text{ from the right. What is the limit?} \]
Trace from the left
Why: Heading for one.
\[ 1 \]
Trace from the right
Why: Heading for three.
\[ 3 \]
Compare
Why: They disagree.
Conclude
Why: No two-sided limit.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
\[ \text{no limit; one-sided limits }1\text{ and }3 \]
Verify: note what does exist
Why: Both one-sided limits exist and are perfectly well defined — the failure is only in their agreement. Saying the limit does not exist without adding that is a less informative answer than the graph supports.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1399-1400
Trap
\[ \text{there is an open circle, so the limit does not exist} \]
Take a missing value as a missing limit
Why: The open circle is read as a failure of the limit.
The curve approaches that height perfectly well from both sides.
An open circle marks a missing value, not a missing limit. The limit ignores the point entirely.
What would make the limit fail is the two sides heading for different heights, which an open circle alone does not indicate.
The circle is exactly the case limits were designed for — behaviour that is clear everywhere except at one point.
Sorting
Circles and dots carry different information.
Sort into buckets
Sort each feature.
Faded example
At a jump discontinuity.
Fill in the blanks
\lim_1f=3 \quad\text___\quad \lim____f=___
Why: The two sides head for different heights, so the two-sided limit fails. Both one-sided limits exist perfectly well, which is the more informative way to describe the situation.
Explain it
Open circles and filled dots mean different things.
Discussion prompt
Explain both to a classmate.
Hint: One is about approaching and one about being.
Answer:
An open circle marks a height the curve approaches but does not attain — the function has no value there, or its value is elsewhere.
A filled dot marks where the function actually is. Both can appear on the same vertical line at different heights, which is how a graph shows a limit and a value disagreeing.
So the open circle is about the limit and the filled dot about the value. A good explanation notes that an open circle alone never prevents a limit — only the two sides disagreeing does that.
Section
Section 4
Concept
A one-sided limit describes what the outputs approach as the inputs come in from below or from above. The two-sided limit exists exactly when both agree.
Reporting both one-sided limits is more useful than reporting that the two-sided limit fails. The failure says only that something is wrong; the two values say exactly what.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1400-1402
Picture it
Two sides heading for different heights.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
Both one-sided limits are perfectly well defined here. The two-sided limit's failure is a statement about their disagreement rather than about either one.
Worked example
More informative than reporting failure.
\[ \text{A function jumps from } 1 \text{ to } 3 \text{ at } x=0. \text{ Describe the limits there.} \]
Find the left-hand limit
Why: Approaching from below.
\[ 1 \]
Find the right-hand limit
Why: Approaching from above.
\[ 3 \]
Compare
Why: They differ.
Report all three facts
Why: Both sides and the failure.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
\[ 1, \; 3, \; \text{no limit} \]
Verify: compare the two reports
Why: Saying only that the limit does not exist discards the two values, which are what describe the function's actual behaviour. The fuller report contains the shorter one and says considerably more.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1400-1401
Prediction
Both one-sided limits exist.
Predict first
What else is needed?
Correct: That they be equal.
Why: The two-sided limit exists exactly when both one-sided limits exist and agree, and then it is their common value. The function's value at the point plays no part.
Worked example
Then the two-sided limit is that common value.
\[ \text{Both one-sided limits at a point equal } 5. \text{ What is the two-sided limit?} \]
Note the left-hand value
Why: Five.
\[ 5 \]
Note the right-hand value
Why: Also five.
\[ 5 \]
Check agreement
Why: They match.
Conclude
Why: The two-sided limit exists.
\[ 5 \]
Figure (svg): The solution to Worked example when the sides agree shown as a ladder of expressions, one row per legal move
\[ \lim=5 \]
Verify: state the relationship
Why: The two-sided limit exists precisely when both one-sided limits exist and are equal, and then it equals their common value. That is the definition rather than a coincidence, which makes it a reliable test.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1401-1402
Error analysis
A student describes a jump discontinuity.
Annotate
On: \( \text{the graph jumps, so no limits exist there} \)
The two-sided limit failing is a statement about disagreement, not about either side being ill-behaved. The two one-sided values are precisely what the jump consists of.
Faded example
Both sides approaching the same value.
Fill in the blanks
\text5=5, \text___=5 \;\Longrightarrow\; \lim=___
Why: Agreement from both directions is exactly what the two-sided limit requires, and its value is the common one. Disagreement would mean no two-sided limit at all.
Sorting
At a jump discontinuity.
Sort into buckets
Sort each item.
Explain it
Saying the limit does not exist is true but thin.
Discussion prompt
Explain what a fuller answer would say.
Hint: What do the two sides do?
Answer:
Both one-sided limits exist and can be stated — the curve heads for a definite height from each direction.
The two-sided limit fails only because those heights disagree, which is a statement about the pair rather than about either one.
So reporting both values plus the disagreement says everything the shorter answer does and considerably more. A good explanation notes that the two values are what the jump actually consists of, so discarding them discards the description.
Section
Section 5
Concept
A limit fails to exist when the two sides disagree, when the outputs grow without bound, or when they oscillate without settling.
The last two rows are the cases where the limit exists, and the second of them is the one that motivated the whole idea. A missing value never prevents a limit; only the surrounding behaviour can.
| the behaviour | what happens |
|---|---|
| the two sides head for different values | a jump; no two-sided limit |
| the outputs grow without bound | no finite limit |
| the outputs oscillate without settling | no limit at all |
| both sides agree on a value | the limit exists |
| the value is missing but the sides agree | the limit still exists |
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1402-1403
Picture it
The commonest of the three failures.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
The other two failures — unbounded growth and oscillation — are about a single side misbehaving rather than the two sides disagreeing, so they fail even one-sidedly.
Worked example
The outputs have nothing to approach.
\[ \text{Does } \lim_{x\to 0}\frac{1}{x^2} \text{ exist?} \]
Check inputs near zero
Why: From both sides.
Check closer inputs
Why: Even nearer.
Note the behaviour
Why: No bound.
Conclude
Why: Nothing is approached.
Figure (svg): The solution to Worked example unbounded growth shown as a ladder of expressions, one row per legal move
\[ \text{no finite limit} \]
Verify: check both sides behave the same
Why: Both sides give large positive values, so this is not a disagreement failure — it is unbounded growth. The two sides agreeing does not help when what they agree on is having no bound.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1402-1403
Sorting
Three failures and two successes.
Sort into buckets
Sort each situation.
Worked example
Three different reasons.
\[ \text{Name the three ways a limit can fail.} \]
First failure
Why: The sides disagree.
Second failure
Why: Growth without bound.
Third failure
Why: Endless oscillation.
Note what does not cause failure
Why: A missing value.
Figure (svg): A graph with a jump, showing different limits approached from the left and from the right
\[ \text{three failures; a missing value is not one} \]
Verify: check the exclusion
Why: A missing value is exactly the situation limits were designed to handle, so it never causes failure. Confusing it with a failure would make the whole idea useless, since holes are the main case of interest.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1403-1403
Trap
\[ \lim_{x\to 0}\frac{1}{x^2}=\infty, \text{ so the limit is infinity} \]
Report the growth as the limit's value
Why: Unbounded behaviour is treated as a number.
Infinity is not a value the outputs approach; it is shorthand for their failing to approach anything.
The limit does not exist in the sense of being a number. The notation is a description of how it fails.
Writing it that way is useful shorthand, provided it is read as 'grows without bound' rather than as an answer.
The distinction matters because a limit is defined as a number the outputs approach, and there is no such number here.
Prediction
A function is undefined at the point being approached.
Predict first
Does the limit fail?
Correct: No; that is the case limits were designed for.
Why: The limit deliberately ignores the point itself, so a missing value there cannot affect it. What causes failure is the surrounding behaviour — disagreement, unbounded growth or oscillation.
Faded example
The outputs at inputs approaching zero.
Fill in the blanks
10^2,\;10^4},\;10^6},\;\ldots \quad\text___
Why: The outputs increase without settling on any value, so there is no number for them to approach. Both sides behaving the same way does not help when neither approaches anything.
Explain it to yourself
A limit can fail in three ways.
Discussion prompt
Explain each and why a missing value is not among them.
Hint: What does a limit require?
Answer:
A limit requires the outputs to approach one number from both sides. It fails if the sides disagree, if the outputs grow without bound, or if they oscillate without settling.
All three are failures of the surrounding behaviour — of what the outputs do as the inputs close in.
A missing value at the point says nothing about that surrounding behaviour, which is why it never causes failure. A good explanation notes that this case is the whole reason limits were invented, so treating it as a failure would defeat the idea entirely.
Comparison
Fill the blanks from memory. Two different questions about the same point.
Comparison matrix
| the value | the limit | |
|---|---|---|
| concerns | the point itself | the inputs near it |
| found by | substituting | examining nearby behaviour |
| can exist without the other | yes | yes |
| affected by the point's value | entirely | not at all |
The last row is the design decision the whole chapter rests on. Ignoring the point is what makes a limit able to describe a hole.
Pattern
Five steps, and the third is what one-sided reasoning misses.
Step 4 is the one that feels wrong and is essential. Substituting the point answers a different question, and the interesting limits are exactly where it fails.
OpenStax Calculus Volume 1, §2.2 The Limit of a Function §2.2
Check
What a limit is.
Check your understanding
What does a limit describe?
Answer: A
Why: A limit concerns behaviour near a point and deliberately ignores the point itself. That is what allows it to exist where the function has no value at all.
Check
One-sided limits.
Check your understanding
The left-hand limit is 2 and the right-hand limit is 5. What is the two-sided limit?
Answer: A
Why: The two-sided limit exists only when both one-sided limits agree. Here they differ, so it fails — though both one-sided limits are perfectly well defined and worth reporting.
Check
Missing values.
Check your understanding
A function is undefined at the point being approached. What follows about the limit?
Answer: A
Why: The limit ignores the point itself, so a missing value there is irrelevant. That case is precisely what limits were designed to handle.
Real world
Instantaneous speed is a limit, and it has to be.
Discussion prompt
A speedometer reads a speed at an instant. Why is that a limit rather than a division?
Hint: What would dividing require?
Answer:
Speed is distance divided by time, but at an instant no time passes and no distance is covered — the division would be zero over zero, which is meaningless.
So instantaneous speed is defined as what the average speed approaches as the time interval shrinks towards zero. That is exactly a limit.
The interval can be made as small as you like without ever being zero, and the averages settle on a value. The whole idea of instantaneous rate depends on limits existing — which is why §12.4 can define a derivative at all.
Commit first
State your confidence along with your answer.
Predict first
Why does the definition of a limit exclude the point itself?
Correct: So it can describe behaviour where the function is undefined.
Why: Including the point would make the limit fail exactly where the idea is most needed — at holes and removed values. Excluding it is what lets a limit say something useful about those cases.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
A classmate computed a limit by substituting the point. Explain when that fails and why.
Hint: What does substitution answer?
Answer:
Substituting gives the function's value, which is a different question from what the outputs approach.
The two agree in ordinary cases, which is why substitution often works. But where the function is undefined there is nothing to substitute, and where the two disagree the answer is simply wrong.
And those are exactly the cases limits exist for. A good explanation notes that when substitution does work, that agreement has a name — continuity — and it is a property to be checked rather than assumed.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The first is the conceptual foundation everything else rests on, and the fourth is what makes the answers precise rather than merely stating that something went wrong.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw a graph with a hole and mark both the limit and the missing value. Beside it, build a two-sided table for the same limit. Underneath, draw a jump and label both one-sided limits and the two-sided failure. Finish by listing the three ways a limit can fail and noting what is not among them.
If your list explicitly excludes a missing value from the failures, the section's central design decision is on the page rather than assumed.
Recap
Five things, and the first is what the rest of the chapter rests on.
| if you remember one thing | it should be this |
|---|---|
| about the definition | it describes what the outputs approach, ignoring the point |
| about tables | sample both sides, and treat the result as evidence |
| about graphs | an open circle marks a missing value, never a missing limit |
| about failure | disagreement, unbounded growth, or oscillation — not a hole |
Section 12.2 replaces estimation with computation, giving the algebraic properties that let most limits be evaluated exactly rather than approximated.
OpenStax, Precalculus, §12.1 Finding Limits: Numerical and Graphical Approaches §12.1, pp. 1388-1403 — everything on these slides traces back here
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