# 1.8 Limits With Infinityap Calculus

Calculus sequences-and-series limits. None of the remaining terms will be negative, so you know this is a bottom limit on what you are going to get. Calculus 8th Edition answers to Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises - Page 70 1 including work step by step written by community members like you. Textbook Authors: Stewart, James, ISBN-10:, ISBN-13: 978-1-28574-062-1, Publisher: Cengage. The corresponding outputs are 8.1, 8.1, 8.01, 8.01, and 8.001. These values are getting closer to 8. The limit of values of f (x) f (x) as x x approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function f (x) = x + 1, x ≠ 7 f (x) = x + 1, x ≠ 7 as x x approaches 7. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.

When working with tables, the best we can do is estimate the limit value.

### Examples

##### Example 1: Using Tables to Estimate Limits

Use the tables shown below to estimate the value of \$\$displaystyle lim_{xto 5} f(x)\$\$.

\$\$ begin{array}{l c} {x} & {f(x)} hline 4.5 & 8.32571hline 4.75 & 8.95692hline 4.9 & 8.99084hline 4.99 & 8.99987hline 4.999 & 8.99992hline 4.9999 & 8.99999hline end{array} \$\$

\$\$ begin{array}{l c} {x} & {f(x)} hline 5.5 & 9.64529hline 5.25 & 9.26566hline 5.1 & 9.04215hline 5.01 & 9.00113hline 5.001 & 9.00011hline 5.0001 & 9.00001hline end{array} \$\$

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach 5...
\$\$ begin{array}{l c} {x} & {f(x)} hline 4.5 & 8.32571hline 4.75 & 8.95692hline 4.9 & 8.99084hline 4.99 & 8.99987hline 4.999 & 8.99992hline 4.9999 & 8.99999hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 9.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach 5...
\$\$ begin{array}{l c} {x} & {f(x)} hline 5.5 & 9.64529hline 5.25 & 9.26566hline 5.1 & 9.04215hline 5.01 & 9.00113hline 5.001 & 9.00011hline 5.0001 & 9.00001hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 9.

Step 3

If the function seems to approach the same value from both directions, then that is the estimate of the limit value.

Answer: \$\$displaystyle lim_{xto 5} f(x) approx 9\$\$.

##### Example 2: Using Tables to Estimate Limits

Using the tables below, estimate \$\$displaystyle lim_{xto-8} f(x)\$\$.

\$\$ begin{array}{l c} {x} & {f(x)} hline -8.5 & 13.1365hline -8.1 & -2.4336hline -8.01 & -2.91313hline -8.001 & -2.99131hline -8.0001 & -2.99913hline -8.00001 & -2.99991hline end{array} \$\$

\$\$ begin{array}{l c} {x} & {f(x)} hline -7.5 & -6hline -7.9 & -5.5hline -7.99 & -5.15hline -7.999 & -5.015hline -7.9999 & -5.0015hline -7.99999 & -5.00015hline end{array} \$\$

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach -8...
\$\$ begin{array}{l c} {x} & {f(x)} hline -8.5 & 13.1365hline -8.1 & -2.4336hline -8.01 & -2.91313hline -8.001 & -2.99131hline -8.0001 & -2.99913hline -8.00001 & -2.99991hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach -3.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach -8...
\$\$ begin{array}{l c} {x} & {f(x)} hline -7.5 & -4hline -7.9 & -3.5hline -7.99 & -3.15hline -7.999 & -3.015hline -7.9999 & -3.0015hline -7.99999 & -3.00015hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach -3.

Step 3

If the function seems to approach different values, then the limit does not exist.

Answer: \$\$displaystyle lim_{xto-8} f(x)\$\$ does not exist.

### Practice Problems

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach 12 from the left...
\$\$ begin{array}{l c} {x} & {f(x)} hline 11.5 & 7hline 11.9 & 7.5hline 11.99 & 7.9hline 11.999 & 7.99hline 11.9999 & 7.999hline 11.99999 & 7.9999hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 8.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach 12 from the right
\$\$ begin{array}{l c} {x} & {f(x))} hline 12.5 & 8.5hline 12.1 & 8.1hline 12.01 & 8.01hline 12.001 & 8.001hline 12.0001 & 8.0001hline 12.00001 & 8.00001hline end{array} \$\$
...f(x) seems to approach 8.

Step 3

If the function seems to approach the same value from both directions, then that is the estimate of the limit value.

\$\$displaystyle lim_{xto 12} f(x) approx 8\$\$.

## 1.8 Limits With Infinityap Calculus Solutions

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach \$\$frac{1}{2}\$\$ from the left...
\$\$ begin{array}{l c} {x} & {f(x)} hline 0 & 1.7hline 0.2 & 1.75hline 0.4 & 1.795hline 0.45 & 1.7995hline 0.49 & 1.79995hline 0.499 & 1.79999hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 1.8.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach \$\$frac{1}{2}\$\$ from the right
\$\$ begin{array}{l c} {x} & {f(x)} hline 1 & -2.44445hline 0.8 & -2.55556hline 0.6 & -2.66667hline 0.55 & -2.77778hline 0.51 & -2.88889hline 0.501 & -2.99999hline end{array} \$\$
\$\$f(x)\$\$ seems to approach -3.

Step 3

If the function seems to approach different values, then the limit does not exist.

\$\$limlimits_{xtofrac{1}{2}} f(x)\$\$ does not exist.

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach 0.75 from the left...
\$\$ begin{array}{l c} {x} & {f(x)} hline 0.7 & 0.1hline 0.72 & -0.01hline 0.74 & 0.001hline 0.749 & -0.0001hline 0.7499 & 0.00001hline 0.74999 & -0.000001hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 0.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach 0.75 from the right...
\$\$ begin{array}{l c} {x} & {f(x)} hline 0.8 & 0.3hline 0.78 & -0.06hline 0.76 & 0.0012hline 0.751 & -0.0006hline 0.7501 & 0.00003hline 0.75001 & -0.000006hline end{array} \$\$
...\$\$f(x)\$\$ seems to approach 0.

Step 3
Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach 10 from the left...
\$\$ begin{array}{l c} {x} & {f(x)} hline 9.5 & 2.3hline 9.9 & 1.8hline 9.99 & 8.3hline 9.999 & 0.8hline 9.9999 & 9.8hline 9.99999 & 2.6hline end{array}; \$\$
...\$\$f(x)\$\$ doesn't seem to approach anything.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right

As the \$\$x\$\$-values approach 10 from the right...
\$\$ begin{array}{l c} {x} & {f(x)} hline 10.5 & 1.1hline 10.1 & 5.8hline 10.01 & 3.6hline 10.001 & 2.9hline 10.0001 & 5.4hline 10.00001 & 12.5hline end{array}; \$\$
...\$\$f(x)\$\$ doesn’t seem to approach anything.

## 1.8 Limits With Infinityap Calculus Solution

Step 3

The function doesn't seem to approach a particular value, so the limit does not exist.

Step 1

Examine what happens as \$\$x\$\$ approaches from the left.

As the \$\$x\$\$-values approach -3 from the left...
\$\$ begin{array}{l c} {x} & {f(x)} hline -4 & 6hline -3.5 & 61hline -3.1 & 611hline -3.01 & 6111hline -3.001 & 61,111hline -3.0001 & 611,111hline end{array} \$\$
...\$\$f(x)\$\$ keeps getting larger.

Step 2

Examine what happens as \$\$x\$\$ approaches from the right.

As the \$\$x\$\$-values approach 3 from the right...
\$\$ begin{array}{l c} {x} & {f(x)} hline -2 & 7hline -2.5 & 72hline -2.9 & 788hline -2.99 & 7656hline -2.999 & 77,701hline -2.9999 & 711,000hline end{array} \$\$
...\$\$f(x)\$\$ keeps getting larger.

Step 3

The function doesn't seem to approach a particular value, so the limit does not exist.