1.4 - Estimating Limit Values from Tables

Limits Through Numerical Analysis

We’ve done limits graphically. Now we’ll find them by looking at numbers in a table. This helps us estimate limits when we can evaluate the function at points very close to (but not equal to) the value we’re approaching.

Recall that when we write:

\lim_{x \to 3} f(x) = 4

it means that as

x

approaches 3 from both sides, the

y

-values approach 4.

Tables of Values

We can approximate limits by evaluating a function at

x

-values that get closer and closer to our target.

Example: basic table analysis

1.4 - Estimating Limit Values from Tables

For the graph above, we could build this table:

$x$	$f(x)$
2.9	3.9
2.99	3.99
2.999	3.999
3	?
3.001	4.001
3.01	4.01
3.1	4.1

x

gets closer to 3 from both sides,

f(x)

approaches 4. So:

\lim_{x \to 3} f(x) = 4

Example: Reading from a Table

Given this table, find

\lim_{x \to -4} g(x)

$x$	$g(x)$
-4.1	2.1
-4.01	2.01
-4.001	2.001
-4	undefined
-3.999	1.999
-3.99	1.99
-3.9	1.9

Looking at values approaching

-4

from both sides: - From the left, values approach 2 - From the right, values approach 2

\lim_{x \to -4} g(x) = 2

Even though

g(-4)

is undefined, the limit still exists because both one-sided approaches agree.

Using a Graphing Calculator

Method 1: table feature

For complicated functions, calculating values by hand gets tedious. Calculators can build tables fast.

Example function:

f(x) = \frac{x^2 + 5x + 6}{x + 2}

Find

\lim_{x \to -2} f(x)

Calculator steps (TI-84 family):

Press Y= and enter the function

Press 2nd then TBLSET (above WINDOW)

Change “Indpnt:” from “Auto” to “Ask”

Press 2nd then TABLE (above GRAPH)

Enter $x$ -values close to $-2$ :

$-2.1$

$-2.01$

$-2.001$

$-2$ (this will show ERROR)

$-1.999$

$-1.99$

$-1.9$

The table shows the corresponding

y

-values.

Limitation: table columns are narrow, so long decimals get rounded or cut off automatically.

Method 2: function notation (more accurate)

This gives more decimal precision.

Enter your function in Y1

Press 2nd then QUIT to get back to the home screen

Press VARS, move to Y-VARS, select Function, choose Y1

Add parentheses and your $x$ -value: Y1(-2.1)

Press ENTER

Time saver: press 2nd then ENTRY to recall the previous calculation and just edit the

x

-value.


Y1(-2.1)        →  0.9
Y1(-2.01)       →  0.99
Y1(-2.001)      →  0.999
Y1(-2)          →  ERROR: Divide by 0
Y1(-1.999)      →  1.001
Y1(-1.99)       →  1.01
Y1(-1.9)        →  1.1

The values approach 1 from both sides, so:

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

Working with Compositions

Sometimes we need limits of composite functions.

Example

Given that

h(x)

is continuous and increasing for all

x \geq 1

, and this table:

$x$	$h(x)$
1.9	4.2
1.95	4.5
1.99	4.8
2	?
2.01	5.2
2.05	5.5
2.1	5.8

Find

\lim_{x \to 2} \sin(h(x))

Step 1: find $\lim_{x \to 2} h(x)$

From the table, as

x \to 2

, the values of

h(x)

approach about 5.

The “continuous and increasing” statement guarantees there are no surprise dips or jumps between the table values.

Step 2: evaluate the outer function

\sin\left(\lim_{x \to 2} h(x)\right) = \sin(5)

Step 3: calculate $\sin(5)$

Make sure your calculator is in RADIAN mode, not degrees. To check, press MODE and confirm “RADIAN” is highlighted.

\sin(5) = -0.958924274\ldots

AP Calculus Decimal Precision Rule

All numerical answers must be accurate to three decimal places. You have two options.

Option 1: rounding

Round to three decimals using standard rules.

For

-0.958924274\ldots

, the fourth decimal is 9, and

9 \geq 5

, so round up. Answer:

-0.959

Option 2: truncating

Just write the first three decimals and stop.

For

-0.958924274\ldots

, take only the first three decimals. Answer:

-0.958

Both are acceptable on the AP exam.

Common calculator answer formats

For a result like

-0.958924274\ldots

Acceptable: -

-0.959

(rounded) -

-0.958

(truncated) -

-0.9589

(extra decimals are fine, graders only check the first three)

Not acceptable: -

-0.96

(only two decimals) -

-0.9

(only one decimal)

AP graders are trained to look at only the first three decimal places. Writing more decimals does nothing for you, since anything past the third is ignored.

Practice Problems

Problem 1

Use the table to find

\lim_{x \to 5} f(x)

$x$	$f(x)$
4.9	7.9
4.99	7.99
4.999	7.999
5.001	8.001
5.01	8.01
5.1	8.1

Answer:

\lim_{x \to 5} f(x) = 8

Problem 2

Given

g(x) = \frac{x^2 - 16}{x - 4}

, use your calculator to estimate

\lim_{x \to 4} g(x)

Suggested

x

-values: 3.9, 3.99, 3.999, 4.001, 4.01, 4.1

$x$	$g(x)$
3.9	7.9
3.99	7.99
3.999	7.999
4.001	8.001
4.01	8.01
4.1	8.1

Answer:

\lim_{x \to 4} g(x) = 8

, since values approach 8 from both sides.

Problem 3

The table shows values for a continuous function

p(x)

$x$	$p(x)$
0.9	2.1
0.95	2.3
0.99	2.7
1.01	3.3
1.05	3.7
1.1	3.9

Find

\lim_{x \to 1} \cos(p(x))

to three decimal places.

From the table,

\lim_{x \to 1} p(x) \approx 3

(values approach 3 from both sides). So:

\cos(3) = -0.989992496\ldots

Answer:

-0.990

(rounded) or

-0.989

(truncated)

Key Takeaways

Tables give numerical evidence for limit values

Choose $x$ -values progressively closer to your target from both sides

The calculator table feature is convenient but can lack precision

The function notation method (Y1(value)) gives more accurate decimals

Always use RADIAN mode for trig functions in calculus

Three decimal places are required for all numerical answers

Statements like “continuous and increasing” guarantee no hidden behavior between table values

Limits can exist even when the function is undefined at that point

For composite functions, work from the inside out

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