Evaluate the function for values of *x* that approach 1 from the left and from the right.

*f*(*x*) = 

9

*x*3 − 1
 
lim *x*→1− *f*(*x*) =  
 
 

lim *x*→1+ *f*(*x*) =

Let's evaluate the function \( f(x) = \frac{9}{x^3 - 1} \) as \( x \) approaches 1 from the left (denoted as \( x \to 1^- \)) and from the right (denoted as \( x \to 1^+ \)).

### 1. Understanding the function:
The function is given by:

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

The expression \( x^3 - 1 \) can be factored using the difference of cubes:

\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]

Thus, the function can be rewritten as:

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

### 2. Evaluating the limit as \( x \to 1^- \):

As \( x \) approaches 1 from the left (values slightly less than 1), the factor \( (x - 1) \) is slightly negative. The factor \( (x^2 + x + 1) \) is always positive for real \( x \).

So, as \( x \) approaches 1 from the left:

- \( (x - 1) \) approaches 0 from the negative side.
- \( (x^2 + x + 1) \) approaches 3 (since \( 1^2 + 1 + 1 = 3 \)).

This makes the function:

\[
f(x) \approx \frac{9}{\text{(a small negative number)} \times 3} \approx -\infty
\]

So,

\[
\lim_{x \to 1^-} f(x) = -\infty
\]

### 3. Evaluating the limit as \( x \to 1^+ \):

As \( x \) approaches 1 from the right (values slightly greater than 1), the factor \( (x - 1) \) is slightly positive. The factor \( (x^2 + x + 1) \) is again always positive.

So, as \( x \) approaches 1 from the right:

- \( (x - 1) \) approaches 0 from the positive side.
- \( (x^2 + x + 1) \) approaches 3.

This makes the function:

\[
f(x) \approx \frac{9}{\text{(a small positive number)} \times 3} \approx +\infty
\]

So,

\[
\lim_{x \to 1^+} f(x) = +\infty
\]

### Final Answer:
- \( \lim_{x \to 1^-} f(x) = -\infty \)
- \( \lim_{x \to 1^+} f(x) = +\infty \)

These results indicate that the function has a vertical asymptote at \( x = 1 \), where the function behavior is sharply different on either side of the asymptote.

Would you like to explore these limits further, or do you have any additional questions?

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Here are 5 related questions for further exploration:

1. How would the function behave as \( x \) approaches 1 from values far from 1?
2. Can you find the limit of \( f(x) \) as \( x \) approaches 0 or infinity?
3. What happens to the function if you change the exponent in the denominator from 3 to 2?
4. How would the limit change if the function were \( \frac{9}{x^3 + 1} \) instead?
5. Can you plot the graph of this function to visually confirm the asymptotic behavior?

**Tip:** When dealing with limits involving asymptotes, always consider the sign and magnitude of the components in the function as they approach critical values.

Understand how to evaluate the limit of the function f(x) = 9 / (x^3 - 1) as x approaches 1 from the left and right. Learn about limits, asymptotic behavior, and rational functions with detailed explanations and examples. Suitable for advanced high school students.

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