We are given the function:

$f(x) = \frac{7}{x^3 - 1}$

We need to evaluate the behavior of this function as $x$ approaches 1 from both the left ($x \to 1^-$) and the right ($x \to 1^+$).

Step 1: Analyze the denominator

The denominator of the function is:

$x^3 - 1$

As $x \to 1$, the denominator becomes:

$1^3 - 1 = 0$

Thus, the denominator approaches zero as $x \to 1$, which indicates a potential vertical asymptote at $x = 1$.

Step 2: Evaluate the limit as $x \to 1^-$ (from the left)

As $x$ approaches 1 from the left, $x^3 - 1$ is slightly negative because $x^3$ is just less than 1. Therefore, the denominator $x^3 - 1$ approaches 0 from the negative side.

Thus, as $x \to 1^-$, $f(x)$ behaves as:

$f(x) \approx \frac{7}{\text{small negative number}} \to -\infty$

So:

$\lim_{x \to 1^-} \frac{7}{x^3 - 1} = -\infty$

Step 3: Evaluate the limit as $x \to 1^+$ (from the right)

As $x$ approaches 1 from the right, $x^3 - 1$ is slightly positive because $x^3$ is just greater than 1. Therefore, the denominator $x^3 - 1$ approaches 0 from the positive side.

Thus, as $x \to 1^+$, $f(x)$ behaves as:

$f(x) \approx \frac{7}{\text{small positive number}} \to +\infty$

So:

$\lim_{x \to 1^+} \frac{7}{x^3 - 1} = +\infty$

Conclusion:

• As $x \to 1^-$, the function approaches $-\infty$.
• As $x \to 1^+$, the function approaches $+\infty$.

This confirms that there is a vertical asymptote at $x = 1$, with the function diverging to negative infinity from the left and positive infinity from the right.

Would you like a detailed breakdown of the steps? If so, feel free to ask!

Here are 5 related questions for further exploration:

1. How do you find the horizontal asymptotes of rational functions?
2. What is the significance of limits in understanding function behavior near asymptotes?
3. Can you use L'Hopital's Rule to solve limits of indeterminate forms?
4. What is the difference between a vertical and a horizontal asymptote?
5. How do you analyze the behavior of polynomial functions near roots?

Tip: Always check both left-hand and right-hand limits when investigating vertical asymptotes!