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.