The problem involves determining the limits of the function $f(x) = 9 + x^{-2}$ as $x$ approaches 0 from the left ($x \to 0^-$), from the right ($x \to 0^+$), and from both directions ($x \to 0$).

The function $f(x) = 9 + x^{-2}$ can be rewritten as:

$f(x) = 9 + \frac{1}{x^2}$

Analysis of the Limits

1. $\lim_{x \to 0^-} f(x)$:

   • When $x$ approaches 0 from the left (negative values of $x$), $x^{-2} = \frac{1}{x^2}$ is still positive because squaring a negative number results in a positive number.
   • However, as $x$ gets closer to 0, $\frac{1}{x^2}$ becomes very large. Hence, $f(x)$ tends to infinity:

   $\lim_{x \to 0^-} \left( 9 + \frac{1}{x^2} \right) = +\infty$

2. $\lim_{x \to 0^+} f(x)$:

   • When $x$ approaches 0 from the right (positive values of $x$), the analysis is similar. $x^{-2} = \frac{1}{x^2}$ is positive and becomes very large as $x$ approaches 0 from the right.
   • Therefore, $f(x)$ also tends to infinity:

   $\lim_{x \to 0^+} \left( 9 + \frac{1}{x^2} \right) = +\infty$

3. $\lim_{x \to 0} f(x)$:

   • Since the limit from the left ($x \to 0^-$) and the limit from the right ($x \to 0^+$) both tend to the same value (+∞), the overall limit as $x$ approaches 0 also exists and equals infinity:

   $\lim_{x \to 0} \left( 9 + \frac{1}{x^2} \right) = +\infty$

Final Answers:

(a) $\lim_{x \to 0^-} f(x) = +\infty$
(b) $\lim_{x \to 0^+} f(x) = +\infty$
(c) $\lim_{x \to 0} f(x) = +\infty$

Do you have any questions or would you like more details? Here are some related questions:

1. What are one-sided limits and how do they differ from regular limits?
2. How does the behavior of $x^{-2}$ affect the limit as $x$ approaches 0?
3. What happens to the function $f(x) = 9 + x^{-2}$ as $x$ becomes very large?
4. How would the limit change if the function were $f(x) = 9 - x^{-2}$?
5. What are some other functions that exhibit similar behavior at certain points?

Tip: When evaluating limits, especially at points where the function might become undefined, it's crucial to consider the behavior from both sides of the point of interest.