Unit 1 · Topic 1.4 · about 30 minutes · 7 parts · 12 scored checks, 3 of them multiple choice

AP Calculus AB 1.4: Estimating Limit Values from Tables

Q: Should I write the estimate as the number the values approach or as the last entry?

The number they approach. In the table for ex - 1x no entry equals 1, but 1 is where both columns are heading, so 1 is the estimate.

Q: Are left-hand and right-hand limits the same thing as one-sided limits?

Yes. A left-hand limit, x c- f(x), uses inputs less than c, and a right-hand limit, x c+ f(x), uses inputs greater than c. Both are one-sided limits.

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Unit 1: 10 to 15% of the multiple-choice section

What you'll be able to do

Estimate a limit, and each one-sided limit, from a table of values of a function.

Key vocabulary

Term	What it means	The exam trap
Limit, $\lim_{x \to c} f (x)$	A real number $R$ such that $f (x)$ can be made arbitrarily close to $R$ by taking $x$ sufficiently close to $c$ , but not equal to $c$ .	Write the arrow and the value $c$ every time; $\lim_{x \to c} f (x)$ names an output, and $c$ is an input.
One-sided limit	A limit taken from one side only: $\lim_{x \to c^{-}} f (x)$ uses inputs less than $c$ , and $\lim_{x \to c^{+}} f (x)$ uses inputs greater than $c$ .	The minus or plus sign names the side the inputs come from, not the sign of the outputs.
Numerical information	Values of a function at chosen inputs, usually given in a table.	A table lists outputs at a few inputs only; it says nothing about the inputs between its rows.
Estimate	A value read from the evidence, here the trend in a table, rather than found exactly.	When a number comes from a table, say so: write "about" or use the word estimate.

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Part 1 Think first How much can compounding add?

Commit to a guess first. Even a wrong one makes the reveal easier to remember.

Predict first

A savings account pays 100 percent interest a year, added in equal pieces. If the year is split into pieces $x$ years long, 1 dollar grows to $(1 + x)^{1 / x}$ dollars by the end of the year. The table shows the amount as the pieces get shorter.

\begin{matrix} x (years) & amount (dollars) \\ 0.1 & 2.59374 \\ 0.01 & 2.70481 \\ 0.001 & 2.71692 \end{matrix}

As $x$ shrinks toward 0, what happens to the amount?

In short. Estimating a limit from a table means reading the outputs at inputs closer and closer to $c$ , on each side, and naming the number they approach.

Rows with $x < c$ estimate the left-hand limit $\lim_{x \to c^{-}} f (x)$ , and rows with $x > c$ estimate the right-hand limit $\lim_{x \to c^{+}} f (x)$ .
When both sides close in on one number, that number estimates $\lim_{x \to c} f (x)$ ; when they approach different numbers or grow without bound, the table points to no limit.
A table samples only a few inputs, so what it gives is an estimate, not a proof.

Part 2 Learn it Three short ideas, each checked as you go

4a. Read the outputs as the inputs close in

A limit asks what $f (x)$ does as $x$ gets close to a number $c$ . A table cannot take $x$ all the way to $c$ , but it can take it close. The inputs $0.9$ , $0.99$ , $0.999$ creep up on 1 from the left, and $1.1$ , $1.01$ , $1.001$ creep down on it from the right.

Here is $f (x) = \frac{e^{x} - 1}{x}$ near $x = 0$ . The table has no row at 0, where the quotient would divide by zero.

Values of f(x) near x = 0
$x$	$f (x)$
-0.1	0.95163
-0.01	0.99502
-0.001	0.99950
0.001	1.00050
0.01	1.00502
0.1	1.05171

Read each side toward 0. From the left the outputs climb: 0.95163, 0.99502, 0.99950. From the right they fall: 1.05171, 1.00502, 1.00050. Both sides close in on 1, and each row is about ten times closer to 1 than the row before it. So the table suggests

\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1.

That number is an estimate: the value the outputs approach, not any one entry. The entry 1.00050 is close to 1, but the limit is the number the whole column is heading for.

Quick check · Calculate

No calculator. The table gives values of a function $g$ near $x = 2$ .

\begin{matrix} x & g (x) \\ 1.9 & 11.41 \\ 1.99 & 11.9401 \\ 1.999 & 11.994001 \\ 2.001 & 12.006001 \\ 2.01 & 12.0601 \\ 2.1 & 12.61 \end{matrix}

Use the table to estimate $\lim_{x \to 2} g (x)$ .

Your answer

4b. One side at a time

A table near $c$ holds two tables: the rows with $x < c$ and the rows with $x > c$ . Each one estimates a one-sided limit. The rows left of $c$ estimate the left-hand limit, $\lim_{x \to c^{-}} f (x)$ , and the rows right of $c$ estimate the right-hand limit, $\lim_{x \to c^{+}} f (x)$ .

When both sides close in on the same number, that number is the estimate of $\lim_{x \to c} f (x)$ . When they do not, the table points to no limit at $c$ :

The two sides close in on different numbers. If the outputs are 2.9, 2.99, 2.999 left of 1 and 0.1, 0.01, 0.001 right of it, the left-hand limit is about 3 and the right-hand limit is about 0, so $\lim_{x \to 1} f (x)$ does not exist.
The outputs grow without bound, such as 100, 10,000 and 1,000,000. They close in on no number.
The outputs keep jumping around as the inputs get closer, and never settle.

An output can approach its number from above on one side and from below on the other. Only the number matters.

Quick check · Sort it

No calculator. Each card lists $f (x)$ at $x = 1.9$ , $1.99$ , $1.999$ (left of 2), then at $x = 2.1$ , $2.01$ , $2.001$ (right of 2). Tap a card, then tap what the table suggests about $\lim_{x \to 2} f (x)$ .

The limit is about 5

No limit: the two sides disagree

No limit: the values are unbounded

4c. The row at c, and what a table can miss

Some tables include a row at $x = c$ itself. Read it, then set it aside. A limit is about inputs close to $c$ but not equal to $c$ , so $f (c)$ can be any number, or not exist at all, without changing the limit. If the outputs near 3 close in on 8 from both sides and the table also says $f (3) = 0$ , the estimate of $\lim_{x \to 3} f (x)$ is still 8. The row at 3 answers a different question: what is $f (3)$ ?

A table is also only a sample. It lists a handful of inputs and says nothing about the ones between them. The function $f (x) = \sin (\frac{π}{x})$ is exactly 0 at $x = \pm 0.1$ , $\pm 0.01$ and $\pm 0.001$ , so a table of those rows looks like it closes in on 0 from both sides. Between those inputs the outputs swing back and forth between $- 1$ and 1, as at $x = \frac{2}{201}$ , where $f (x) = 1$ , and the limit at 0 does not exist.

So read rows on both sides, closer and closer to $c$ , and call what you find an estimate.

Quick check · Multiple choice

No calculator. The table gives values of a function $h$ near $x = - 1$ .

\begin{matrix} x & h (x) \\ - 1.1 & 2.3 \\ - 1.01 & 2.03 \\ - 1.001 & 2.003 \\ - 1 & 6 \\ - 0.999 & 1.997 \\ - 0.99 & 1.97 \\ - 0.9 & 1.7 \end{matrix}

Which is the best estimate of $\lim_{x \to - 1} h (x)$ ?

Common mistakes

The mistake	Why it is wrong	Do this instead
Reading the row at $x = c$ as the limit.	The limit depends on inputs near $c$ , never on $c$ itself, so $f (c)$ can differ from the limit or not exist at all.	Note $f (c)$ , then set its row aside and read the rows on each side of $c$ .
Reading only one side of the table.	A two-sided limit needs both one-sided limits to exist and to be equal, and one side alone cannot show that.	Estimate the left-hand and right-hand limits separately, then compare them.
Giving the entry closest to $c$ as the limit.	Every entry is an output at an input that is not $c$ . The limit is the number the entries approach.	Look at how the entries change from row to row, and give the number they are closing in on.

Part 3 Watch it done A limit read from both sides

Watch each step, then do the last one yourself.

The table gives values of $f (x) = \frac{\sqrt{x + 4} - 2}{x}$ near $x = 0$ . Use it to estimate $\lim_{x \to 0} f (x)$ .

\begin{matrix} x & f (x) \\ - 0.1 & 0.251582 \\ - 0.01 & 0.250156 \\ - 0.001 & 0.250016 \\ 0.001 & 0.249984 \\ 0.01 & 0.249844 \\ 0.1 & 0.248457 \end{matrix}

Step 1 · Split the table at 0
The rows $x = - 0.1$ , $- 0.01$ , $- 0.001$ approach 0 from the left, and $x = 0.1$ , $0.01$ , $0.001$ approach it from the right. There is no row at 0, where the quotient is $\frac{0}{0}$ .
Step 2 · Read the left side
In order toward 0: 0.251582, 0.250156, 0.250016. The outputs fall toward 0.25, and their distance from it shrinks from about 0.0016 to 0.00016 to 0.000016. So $\lim_{x \to 0^{-}} f (x)$ is about 0.25.
Step 3 · Read the right side
In order toward 0: 0.248457, 0.249844, 0.249984. The outputs rise toward 0.25 from below, with gaps shrinking the same way. So $\lim_{x \to 0^{+}} f (x)$ is about 0.25.
Why this step?
Why read the two sides separately instead of only the two rows nearest 0?
A two-sided limit exists only when both one-sided limits exist and agree. The two nearest entries, 0.250016 and 0.249984, are close to each other, but a pair of entries cannot show where each side is heading. A whole column on each side can.
Your step · Calculate

No calculator. Finish it. Both one-sided estimates for $f (x) = \frac{\sqrt{x + 4} - 2}{x}$ are about 0.25. What is the best estimate of $\lim_{x \to 0} f (x)$ ? Give a decimal.

Your answer

Part 4 Try it Two new tables

Try it · Decide and justify

No calculator. A table gives only three values: $p (0.1) = 3.31$ , $p (0.01) = 3.0301$ and $p (0.001) = 3.003001$ . A classmate says the table shows that $\lim_{x \to 0} p (x)$ is about 3. Is the classmate right?

Which reason supports your answer?

Try it · Calculate

No calculator. A switch in a circuit closes at time $t = 0$ seconds. The table gives the current $I (t)$ , in amperes, at times near 0.

\begin{matrix} t (seconds) & I (t) (amperes) \\ - 0.1 & 0 \\ - 0.01 & 0 \\ - 0.001 & 0 \\ 0.001 & 2.994 \\ 0.01 & 2.9406 \\ 0.1 & 2.4562 \end{matrix}

Use the table to estimate $\lim_{t \to 0^{+}} I (t)$ , in amperes.

Your answer

Part 5 Explain it Write it the way the exam asks

Free response is written by hand on the exam, so answer on paper first. Nothing you type here leaves this page. Then open the model answer and tick each point you earned.

Free response is 50% of the exam score. This one is shaped like Question 5, from free-response Part B (calculator not permitted).

Free-response practice · Question 5 style

No calculator. A cart moves along a straight track. For $h \neq 0$ , $A (h)$ is the cart's average velocity, in meters per second, over the time interval from $t = 3$ to $t = 3 + h$ seconds. A second cart is bumped at $t = 3$ , and $B (h)$ is its average velocity, in meters per second, over the same interval. Selected values are in the table.

\begin{matrix} h & A (h) & B (h) \\ - 0.1 & 2.0127 & 1.525 \\ - 0.01 & 2.0013 & 1.5025 \\ - 0.001 & 2.0001 & 1.50025 \\ 0.001 & 1.9999 & 2.5005 \\ 0.01 & 1.9988 & 2.505 \\ 0.1 & 1.9877 & 2.55 \end{matrix}

(a) Use the table to estimate $\lim_{h \to 0^{-}} A (h)$ and $\lim_{h \to 0^{+}} A (h)$ . Explain your reasoning.

(b) Use your answers from part (a) to estimate $\lim_{h \to 0} A (h)$ . Using correct units, interpret the meaning of this limit in the context of the problem.

Your answer stays on this page. It is never sent or saved.

Model answer

(a) For $h = - 0.1$ , $- 0.01$ , $- 0.001$ the values of $A (h)$ are 2.0127, 2.0013, 2.0001, closing in on 2, so $\lim_{h \to 0^{-}} A (h)$ is about 2. For $h = 0.1$ , $0.01$ , $0.001$ they are 1.9877, 1.9988, 1.9999, also closing in on 2, so $\lim_{h \to 0^{+}} A (h)$ is about 2.

(b) Both one-sided estimates are 2, so $\lim_{h \to 0} A (h)$ is about 2. The average velocities over shorter and shorter intervals with one end at $t = 3$ approach 2, so at the instant $t = 3$ seconds the cart's velocity is about 2 meters per second.

(c) No. For $h < 0$ the values of $B (h)$ , 1.525, 1.5025, 1.50025, close in on 1.5, and for $h > 0$ the values 2.55, 2.505, 2.5005 close in on 2.5. The one-sided limits appear to be 1.5 and 2.5, which are not equal, so the table does not support the claim that $\lim_{h \to 0} B (h)$ exists.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Reads the rows with $h < 0$ and estimates $\lim_{h \to 0^{-}} A (h)$ as 2.
Reads the rows with $h > 0$ and estimates $\lim_{h \to 0^{+}} A (h)$ as 2.
Estimates $\lim_{h \to 0} A (h)$ as 2 because the one-sided estimates agree.
Interprets the limit as the cart's velocity at the instant $t = 3$ seconds, about 2 meters per second, with units.
Answers no for $B$ , naming the left-hand estimate 1.5 and the right-hand estimate 2.5 and saying they are not equal.

Part 6 Check yourself A table with a gap at 3

Both case questions use the same table. Answer them from it.

The table gives values of a function $r$ near $x = 3$ , including the value at 3 itself.

Values of r near x = 3
$x$	$r (x)$
2.9	0.7
2.99	0.97
2.999	0.997
3	5
3.001	4.003
3.01	4.03
3.1	4.3

Case question 1 · Fill the statement

No calculator. Tap a blank, then tap an answer.

From the table, $\lim_{x \to 3^{-}} r (x)$ is about and $\lim_{x \to 3^{+}} r (x)$ is about .

Case question 2 · Multiple choice

No calculator. Which statement about $\lim_{x \to 3} r (x)$ is best supported by the table?

The rest stand on their own and do not use the case.

Check yourself · Spot the error

No calculator. A student uses a table to decide whether $\lim_{x \to 3} g (x)$ exists. Tap the first line with a mistake.

Check yourself · Match

No calculator. A table gives $f (x)$ at $x = 3.9$ , $3.99$ , $3.999$ , $4$ , $4.001$ , $4.01$ and $4.1$ . Tap each quantity, then the rows you read to estimate or find it.

Part 7 Keep it Bring back earlier ideas

Reading a limit from a table rests on two earlier ideas in this unit. Pulling them back up is what makes them last.

Bring it back · Multiple choice

No calculator. On the graph of $g$ , as $x$ approaches 1 from the left the graph approaches the open circle at $(1, 3)$ , and as $x$ approaches 1 from the right it approaches the closed dot at $(1, - 1)$ . Which statement is true?

Bring it back · Calculate

No calculator. For every $h \neq 0$ , the average velocity of a ball over the time interval from $t = 1$ to $t = 1 + h$ seconds is $15 - 5 h$ meters per second. What is the ball's velocity, in meters per second, at the instant $t = 1$ ?

Your answer

In three lines

Rows on each side of c estimate the one-sided limits: the number the outputs approach as the inputs close in.
When both sides approach the same number, that number estimates the limit; when they differ or grow without bound, there is no limit.
A row at c gives the value of the function there, which is not part of the limit, and a table only ever gives an estimate.

Exam strategy for this topic

Section I Part A, Multiple Choice (calculator not permitted): 29 questions in 62 minutes.
Section I Part B, Multiple Choice (graphing calculator required): 13 questions in 38 minutes.
Section II Part A, Free Response (graphing calculator required): 2 questions in 30 minutes.
Section II Part B, Free Response (calculator not permitted): 4 questions in 60 minutes.

When a question gives a table, say which rows you read: name the inputs on each side of c and the number the outputs approach there. Give each one-sided estimate before the two-sided one, and when the two sides disagree, name both values and say they are not equal.

Questions students ask

How close to

c

do the inputs in a table need to be?

Close enough that the pattern on each side is clear. Inputs that move ten times closer each row, such as $0.1$ , $0.01$ and $0.001$ away from $c$ , usually make the trend plain. If the rows still disagree, look at inputs closer still.

Can a table prove that a limit has a certain value?

No. A table shows a few inputs, so it supports an estimate. The function $\sin (\frac{π}{x})$ is 0 at $x = \pm 0.1$ , $\pm 0.01$ and $\pm 0.001$ but has no limit at 0. Exact values come from algebra and the properties of limits, which start in the next topic.

Do the values have to approach from the same direction on both sides?

No. The outputs can rise toward the number on one side and fall toward it on the other, as in the table for $\frac{e^{x} - 1}{x}$ . What matters is that both sides close in on the same number.

Should I write the estimate as the number the values approach or as the last entry?

The number they approach. In the table for $\frac{e^{x} - 1}{x}$ no entry equals 1, but 1 is where both columns are heading, so 1 is the estimate.

Are left-hand and right-hand limits the same thing as one-sided limits?

Yes. A left-hand limit, $\lim_{x \to c^{-}} f (x)$ , uses inputs less than $c$ , and a right-hand limit, $\lim_{x \to c^{+}} f (x)$ , uses inputs greater than $c$ . Both are one-sided limits.

Course alignment, for teachers

AP Calculus AB topic 1.4, Unit 1: Limits and Continuity.

Skill 2.B: Identify mathematical information from graphical, numerical, analytical, and/or verbal representations.

This lesson maps the one essential knowledge statement of the topic to the part that teaches each one: 1 is checked by a scored question. The map is in the lesson file.