Let's first analyze the function:

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

We are interested in evaluating the left-hand limit $\lim_{x \to 1^-} f(x)$ and the right-hand limit $\lim_{x \to 1^+} f(x)$. To do this, we'll examine the behavior of $f(x)$ near $x = 1$ by substituting values approaching 1 from the left (e.g., 0.5, 0.9, 0.99, etc.) and from the right (e.g., 1.5, 1.1, 1.01, etc.).

Function Breakdown:

The function can be rewritten as: $f(x) = \frac{9}{(x-1)(x^2+x+1)}$ Thus, $f(x)$ has a discontinuity at $x = 1$ because the denominator approaches zero as $x$ approaches 1.

For the Left-Hand Limit $x \to 1^-$:

• As $x \to 1^-$, $x - 1$ becomes a small negative number, and $(x^2 + x + 1)$ is positive, so the overall expression for $f(x)$ tends to negative infinity.

For the Right-Hand Limit $x \to 1^+$:

• As $x \to 1^+$, $x - 1$ becomes a small positive number, and $(x^2 + x + 1)$ remains positive, so the overall expression for $f(x)$ tends to positive infinity.

Now let's use the table values provided to evaluate the function near $x = 1$:

x	f(x)
0.5	-10.3
0.9	-33.2
0.99	-303.0
0.999	-3003.0
0.9999	-30003.0
1.0001	299997
1.001	29997.0
1.01	2997.0
1.1	297.0
1.5	27.2

From this table, we observe the following:

• As $x$ approaches 1 from the left (values like 0.9, 0.99, 0.999), the values of $f(x)$ become very large negative numbers, indicating the left-hand limit tends to negative infinity.
• As $x$ approaches 1 from the right (values like 1.1, 1.01, 1.0001), the values of $f(x)$ become very large positive numbers, indicating the right-hand limit tends to positive infinity.

Thus:

• $\lim_{x \to 1^-} f(x) = -\infty$
• $\lim_{x \to 1^+} f(x) = +\infty$

Would you like to explore more details, or have any further questions on this?

Follow-up questions:

1. How would the graph of $f(x) = \frac{9}{x^3 - 1}$ behave near $x = 1$?
2. What happens if we take the limit of $f(x)$ as $x$ approaches 1 from different directions, say from $x = 2$ or $x = 0$?
3. Can we apply L'Hopital's Rule to evaluate the limit as $x \to 1$?
4. How would the behavior change if the function were $f(x) = \frac{1}{x^3 - 1}$ instead of $\frac{9}{x^3 - 1}$?
5. What is the role of the cubic term $x^3$ in determining the limits?

Tip:

Always carefully consider the sign of the denominator as $x$ approaches the point where the function is undefined. It helps determine whether the function tends to $+\infty$ or $-\infty$.