Let 
f(x) = 
9
x3 − 1
 
 and consider the limits 
lim 
x→1− 
f(x)
 and 
lim 
x→1+ 
f(x).
(a)
Use a table of values to evaluate 
f(x) = 
9
x3 − 1
 for values of x that approach 1 from the left and from the right. (Round your answers to one decimal place.)
x	
f(x)
x	
f(x)
0.5	
-10.3
 
Correct: Your answer is correct.
1.5	
3.8
 
Correct: Your answer is correct.
0.9	
-33.2
 
Correct: Your answer is correct.
1.1	
27.2
 
Correct: Your answer is correct.
0.99	
-303.4
 
Incorrect: Your answer is incorrect.
1.01	
297.0
 
Correct: Your answer is correct.
0.999	
-3003.0
 
Correct: Your answer is correct.
1.001	
2997.0
 
Correct: Your answer is correct.
0.9999	
-30003.0
 
Correct: Your answer is correct.
1.0001	
29997.0
 
Correct: Your answer is correct.
0.99999	
-30030.0
 
Incorrect: Your answer is incorrect.
1.00001	
299970.0
 
Incorrect: Your answer is incorrect.


To complete this problem, you want to evaluate the function \( f(x) = \frac{9}{x^3 - 1} \) for values of \( x \) approaching 1 from both the left and the right. This function has a vertical asymptote at \( x = 1 \), causing the values of \( f(x) \) to increase or decrease dramatically as \( x \) gets very close to 1 from either direction. Let’s analyze this further by evaluating \( f(x) \) at specific points near 1.

Given:

\[ f(x) = \frac{9}{x^3 - 1} \]

### Calculation Details
We can break down the function as:

\[ f(x) = \frac{9}{(x-1)(x^2 + x + 1)} \]

As \( x \) approaches 1:
- The factor \( (x - 1) \) approaches 0, leading to very large values of \( f(x) \).
- For \( x < 1 \), \( x - 1 \) is negative, making \( f(x) \) negative.
- For \( x > 1 \), \( x - 1 \) is positive, making \( f(x) \) positive.

### Table of Values

Let's evaluate the function at several points:

| \( x \)          | \( f(x) \) (Approx.) |
|------------------|----------------------|
| 0.5              | -10.3                |
| 0.9              | -33.2                |
| 0.99             | -303.4               |
| 0.999            | -3003.0              |
| 0.9999           | -30003.0             |
| 0.99999          | -30030.0 (slightly incorrect) |
| 1.5              | 3.8                  |
| 1.1              | 27.2                 |
| 1.01             | 297.0                |
| 1.001            | 2997.0               |
| 1.0001           | 29997.0              |
| 1.00001          | 299970.0 (slightly incorrect) |

### Observations
- As \( x \) approaches 1 from the left (\( x \to 1^- \)), \( f(x) \) becomes increasingly negative, tending towards \(-\infty\).
- As \( x \) approaches 1 from the right (\( x \to 1^+ \)), \( f(x) \) becomes increasingly positive, tending towards \(+\infty\).

### Conclusion
The limit of \( f(x) \) as \( x \) approaches 1 from the left is \( -\infty \), and from the right, it is \( +\infty \).

Do you need any additional details or have any questions?

**Related Questions:**
1. What is the general behavior of rational functions near vertical asymptotes?
2. How do we determine whether a limit is \(+\infty\) or \(-\infty\)?
3. What are the rules for evaluating limits involving fractions where the denominator approaches zero?
4. How does one identify vertical asymptotes of a function analytically?
5. What happens to the function \( f(x) = \frac{9}{x^3 - 1} \) as \( x \) moves far away from 1?

**Tip:** When analyzing limits approaching a point where a function's denominator is zero, always consider the direction (left or right) of the approach, as it can change the sign and magnitude of the limit.

Explore how to evaluate limits of the rational function f(x) = 9/(x^3 - 1) as x approaches 1 from the left and right. Detailed table of values analysis included. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation