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We are tasked with determining why the function \( f(x) = \frac{1}{x + 5} \) is discontinuous at \( a = -5 \). Let's analyze the situation step by step:

1. **Function at \( x = -5 \):**
   \[
   f(-5) = \frac{1}{-5 + 5} = \frac{1}{0}
   \]
   Since division by zero is undefined, \( f(-5) \) does not exist.

2. **Left-hand and Right-hand Limits:**
   - **Left-hand limit (\( \lim_{x \to -5^-} f(x) \)):**
     Approaching from the left side of \( -5 \), values of \( x + 5 \) are slightly negative. Thus:
     \[
     \lim_{x \to -5^-} f(x) = \lim_{x \to -5^-} \frac{1}{x + 5} = -\infty
     \]
   
   - **Right-hand limit (\( \lim_{x \to -5^+} f(x) \)):**
     Approaching from the right side of \( -5 \), values of \( x + 5 \) are slightly positive. Thus:
     \[
     \lim_{x \to -5^+} f(x) = \lim_{x \to -5^+} \frac{1}{x + 5} = +\infty
     \]

3. **Overall Limit:** 
   Since the left-hand and right-hand limits are not equal, the limit of the function as \( x \to -5 \) does not exist.

### Correct Answers:
- \( f(-5) \) is undefined.
- \( \lim_{x \to -5} f(x) \) does not exist.

These match the statements in the options: 
- "f(−5) is undefined."
- "lim \(x \to -5\) \(f(x)\) does not exist."

### Final Answer:
The correct selections are:
- \( f(-5) \) is undefined.
- \( \lim_{x \to -5} f(x) \) does not exist.

Would you like more details or have any questions about this?

Here are 5 related questions to expand on this:
1. How do you determine if a function is continuous at a given point?
2. What is the difference between one-sided and two-sided limits?
3. Why does the limit of a function not exist if the left-hand and right-hand limits are different?
4. What happens when a function has a vertical asymptote?
5. How do you deal with limits that approach infinity?

**Tip:** Always check if the function is defined at the point in question before checking the limit for continuity.

Explain why the function is discontinuous at the given number a. (Select all that apply.) f(x) = 1 / (x + 5), a = -5

Learn why the function f(x) = 1 / (x + 5) is discontinuous at a = -5. This explanation covers key mathematical concepts such as limits, continuity, and why the function's value and limit do not exist at the point of discontinuity. Suitable for high school students.

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