Why should I not just plug in $x = a$?

Because the limit is about the values near $a$, not at $a$. Often $f(a)$ is undefined (a hole or $\frac{0}{0}$), and sometimes it exists but differs from the limit. The table shows the trend that $f(a)$ cannot show.

Is a table a proof that the limit is correct?

No. A table only gives strong evidence. Exact answers come from algebra (Limits by Direct Substitution, factoring) or from limit laws. Tables are excellent for estimating and for checking an algebraic answer.

What does it mean when my calculator shows strange values very close to $a$?

With extremely small differences like $x - a = 0.0000000001$, a calculator can lose digits when it subtracts nearly equal numbers. If the values suddenly jump around, step back to $a \pm 0.001$ or $a \pm 0.0001$, where the arithmetic is still reliable.

Teacher Resources

Evaluating Limits from Tables

Q: How many values do I need in the table?

Usually three on each side, such as $a \pm 0.1$, $a \pm 0.01$, $a \pm 0.001$, are enough to see a clear trend. If the trend is not clear, add values that are even closer, like $a \pm 0.0001$.

Estimate a limit numerically: build a table of values from both sides, move closer and closer, and read where the outputs are heading.

Advanced30 minLesson

Printable worksheet

Definition

To evaluate a limit from a table, you choose

x

values that get closer and closer to

a

from both sides, compute

f(x)

for each one, and look at the value the outputs approach.

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

How to build the table:

Left side ( $x < a$ ): for example $a - 0.1$ , $a - 0.01$ , $a - 0.001$
Right side ( $x > a$ ): for example $a + 0.001$ , $a + 0.01$ , $a + 0.1$
Skip $x = a$ itself: the limit does not care about $f(a)$

Example:

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

is undefined at

x = 2

, but the table shows where it is heading:

$x$	$1.9$	$1.99$	$1.999$	$2.001$	$2.01$	$2.1$
$f(x)$	$3.9$	$3.99$	$3.999$	$4.001$	$4.01$	$4.1$

Both sides approach

4

, so we estimate

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4

Three possible outcomes:

Both sides approach the same number $L$ : the limit is $L$
The sides approach different numbers: the limit does not exist (DNE)
The outputs grow without bound: we write $\infty$ (no finite limit)

A table gives an estimate, not a proof. It is the numerical side of the same idea you saw in Evaluating Limits Graphically.

Try it now

You want to estimate

\lim_{x \to 3} f(x)

with a table of values. Which set of

x

values is the best choice?

Worked Examples

Use a table to estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ .

Check the point itself

At $x = 2$ : $\frac{4 - 4}{2 - 2} = \frac{0}{0}$ → $f(2)$ is undefined, so we need a table

Approach from the left

$f(1.9) = 3.9$ , $f(1.99) = 3.99$ , $f(1.999) = 3.999$ → Left side heads toward $4$

Approach from the right

$f(2.1) = 4.1$ , $f(2.01) = 4.01$ , $f(2.001) = 4.001$ → Right side heads toward $4$

Compare both sides

Both sides approach the same value → $\lim_{x \to 2} f(x) = 4$

Answer: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$ , even though $f(2)$ does not exist

Common Mistakes

Using only one side of $a$

Why it's wrong: A table with only $x = 1.9$ , $1.99$ , $1.999$ shows the left-hand limit. The right side may head somewhere else, as with $\frac{|x|}{x}$ .

Correct: Always build both columns: values just below $a$ and values just above $a$ .

Choosing $x$ values that are not close to $a$

Why it's wrong: Values like $x = 1$ , $1.5$ , $2.5$ , $3$ are too far away. The function can behave very differently far from the point.

Correct: Move closer by a factor of $10$ each time: $a \pm 0.1$ , $a \pm 0.01$ , $a \pm 0.001$ .

Putting $x = a$ in the table and using $f(a)$ as the limit

Why it's wrong: The limit describes what happens near $a$ , not at $a$ . At a hole $f(a)$ may be undefined, and at a moved point it may be a different number.

Correct: Leave $x = a$ out of the table and read only the trend of the nearby values.

Trusting a few values blindly

Why it's wrong: Special choices of $x$ can hide oscillation, as with $\sin\left(\frac{\pi}{x}\right)$ , and rounding can hide small differences between the sides.

Correct: Use several values that are not all of the same pattern, keep enough decimals, and confirm with a graph or algebra.

Interactive Visual

Function graph

Tap or hover over the graph to read off values.

f(x) = (x² - 4)/(x - 2)

Table of values

x	1.9	1.99	1.999	2.001	2.01	2.1
y	3.9	3.99	3.999	4.001	4.01	4.1

Function graph

Tap or hover over the graph to read off values.

g(x) = |x|/x

At x = 0: from the left g(x) → -1, from the right g(x) → 1. The values differ, so the limit does not exist (jump). g(0) is undefined.

Table of values

x	-0.1	-0.01	-0.001	0.001	0.01	0.1
y	-1	-1	-1	1	1	1

Function graph

Tap or hover over the graph to read off values.

h(x) = 1/(x - 3)²Asymptote

Table of values

x	2.9	2.99	2.999	3.001	3.01	3.1
y	100	10000	1000000	1000000	10000	100

Interactive Sandbox

Interactive Grapher

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

You want to estimate $\lim_{x \to 3} f(x)$ with a table of values. Which set of $x$ values is the best choice?

Why It Matters

Tables are often the first and most practical way to find a limit:

No formula trick needed: functions like $\frac{\sin x}{x}$ or $\frac{e^x - 1}{x}$ cannot be simplified by factoring, but a table finds their limits in seconds
Calculators and computers estimate limits exactly this way: they plug in numbers closer and closer to the point
Checking your algebra: if your algebra says the limit is $4$ but the table heads toward $3$ , something went wrong
Calculus foundations: instantaneous speed and slope of a curve start as tables of average speeds over smaller and smaller time intervals

Knowing when a table can fool you (oscillating functions, rounding) is just as important as building one.

Real World Applications

Instantaneous Speed in Physics

A speedometer shows speed at one instant. Physicists find it by computing average speeds over shorter and shorter time intervals and reading the trend in the table.

Example:

A ball falls $s(t) = 5t^2$ meters in $t$ seconds. Average speeds starting at $t = 2$ : over $0.1$ s it is $20.5$ m/s, over $0.01$ s it is $20.05$ m/s, over $0.001$ s it is $20.005$ m/s.

1Try It Yourself

The average speed of the ball between $t = 2$ and $t = 2 + h$ seconds is $\frac{5(2 + h)^2 - 20}{h}$ m/s.

What is the instantaneous speed at $t = 2$ seconds, in m/s?

Step 1: Write the mathematical expression

Make $h$ smaller and smaller and read where the average speed is heading:

Growth Rates in Biology

A bacteria culture that doubles every hour does not grow in jumps. Its instantaneous growth rate is found from a table of the expression $\frac{2^h - 1}{h}$ for tiny time steps $h$.

Example:

For $h = 0.1$ hours the table gives $0.7177$ , for $h = 0.01$ it gives $0.6956$ , for $h = 0.001$ it gives $0.6934$ : the rate settles near $0.693$ per hour.

2Try It Yourself

Biologists estimate $\lim_{h \to 0} \frac{2^h - 1}{h}$ with a table. Use a calculator.

What is the limit, rounded to two decimal places?

Step 1: Write the mathematical expression

Compute the expression for smaller and smaller $h$ :

Signal Processing and Audio

Audio software uses functions like $\frac{\sin(2x)}{x}$ to filter sound. At $x = 0$ the formula gives $\frac{0}{0}$, so programmers use the limit from a table to fill in the missing value.

Example:

Near $0$ (radians), $\frac{\sin(2x)}{x}$ gives $1.98669$ at $x = 0.1$ and $1.99987$ at $x = 0.01$ .

3Try It Yourself

A programmer needs the value that $\frac{\sin(2x)}{x}$ approaches as $x \to 0$ ( $x$ in radians).

What value should the program use at $x = 0$ ?

Step 1: Write the mathematical expression

Build a table with $x = 0.1$ , $0.01$ , $0.001$ :

Key Takeaways

1To estimate $\lim_{x \to a} f(x)$ from a table, use $x$ values closer and closer to $a$ from both sides
2A good pattern: $a \pm 0.1$ , $a \pm 0.01$ , $a \pm 0.001$ ; leave $x = a$ out
3If both sides approach the same number $L$ , the limit is $L$ , even when $f(a)$ is undefined
4If the sides approach different numbers, the limit does not exist
5If the outputs grow without bound, there is no finite limit and we write $\infty$ (or $-\infty$ )
6A table is an estimate: oscillating functions like $\sin\left(\frac{\pi}{x}\right)$ and rounding can fool it, so confirm with a graph or algebra

Frequently Asked Questions

Usually three on each side, such as

a \pm 0.1

a \pm 0.01

a \pm 0.001

, are enough to see a clear trend. If the trend is not clear, add values that are even closer, like

a \pm 0.0001

Glossary

Table of values: A list of chosen inputs $x$ and their outputs $f(x)$ , used to see a trend
Left-hand limit: The value $f(x)$ approaches as $x \to a$ through values smaller than $a$ , written $\lim_{x \to a^-} f(x)$
Right-hand limit: The value $f(x)$ approaches as $x \to a$ through values larger than $a$ , written $\lim_{x \to a^+} f(x)$
Does not exist (DNE): The outputs do not approach one single finite number, for example because the two sides disagree or the function oscillates
Grows without bound: The outputs become larger than any number you choose; written as a limit equal to $\infty$
Oscillation: Behavior where the outputs keep swinging between values instead of settling down, like $\sin\left(\frac{\pi}{x}\right)$ near $0$

Formula Card

Table pattern

x = a - 0.1,\ a - 0.01,\ a - 0.001 \ \mid\ a + 0.001,\ a + 0.01,\ a + 0.1

Approach $a$ from both sides, ten times closer each step, and never use $x = a$ itself

Limit exists

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

Both columns of the table head toward the same number

Limit does not exist

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

The two columns approach different numbers, or the values oscillate

Evaluating Limits from Tables

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Table of values

Function graph

Table of values

Function graph

Table of values

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Instantaneous Speed in Physics

Growth Rates in Biology

Signal Processing and Audio

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Table pattern

Limit exists

Limit does not exist

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits Graphically

Limits by Direct Substitution