Use the graph of the function f to state the value of each limit, if it exists. (If an answer does not exist, enter DNE.)
f(x) = x

9 + x−2
(a)
lim
x→0−
f(x)

Incorrect: Your answer is incorrect.
(b)
lim
x→0+
f(x)

Incorrect: Your answer is incorrect.
(c)
lim
x→0

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.

Learn how to evaluate the limits of the function f(x) = 9 + x^-2 as x approaches 0 from the left, right, and both sides. Detailed explanation with examples and analysis provided. Suitable for college-level mathematics.

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