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


f(−5) and 
lim 
x→−5 
f(x) are finite, but are not equal.

f(−5) is undefined.

lim 
x→−5+ 
 f(x) and 
lim 
x→−5− 
 f(x) are not finite, and are not equal.

lim 
x→−5 
 f(x) does not exist.

none of the above

To determine why the function \( f(x) = \frac{1}{x+5} \) is discontinuous at \( a = -5 \), let's analyze the function's behavior at that point.

### Step 1: Is \( f(-5) \) defined?
The function \( f(x) \) is given as \( f(x) = \frac{1}{x+5} \). To evaluate \( f(-5) \), substitute \( x = -5 \) into the function:
\[
f(-5) = \frac{1}{-5+5} = \frac{1}{0}
\]
Since division by zero is undefined, \( f(-5) \) is **undefined**.

### Step 2: Does the limit \( \lim_{x \to -5} f(x) \) exist?
We must check the left-hand limit \( \lim_{x \to -5^-} f(x) \) and the right-hand limit \( \lim_{x \to -5^+} f(x) \).

- As \( x \to -5^- \) (approaching -5 from the left), \( x+5 \) is slightly negative, making \( \frac{1}{x+5} \) approach negative infinity: 
  \[
  \lim_{x \to -5^-} f(x) = -\infty
  \]

- As \( x \to -5^+ \) (approaching -5 from the right), \( x+5 \) is slightly positive, making \( \frac{1}{x+5} \) approach positive infinity:
  \[
  \lim_{x \to -5^+} f(x) = +\infty
  \]

Since these two one-sided limits are not equal (and are not finite), the overall limit \( \lim_{x \to -5} f(x) \) **does not exist**.

### Conclusion
Based on the analysis, the correct reasons why the function is discontinuous at \( a = -5 \) are:

- **\( f(-5) \) is undefined.**
- **\( \lim_{x \to -5^+} f(x) \) and \( \lim_{x \to -5^-} f(x) \) are not finite, and are not equal.**

So the correct selections are:
- "f(-5) is undefined."
- "lim \( x \to -5^+ \) f(x) and lim \( x \to -5^- \) f(x) are not finite, and are not equal."

Would you like more details or have any further questions?

---

Here are 5 related questions to explore:
1. What are the types of discontinuities in functions?
2. How do you determine if a function is continuous at a given point?
3. What is the difference between a removable and non-removable discontinuity?
4. How do you compute one-sided limits for rational functions?
5. What happens if the left-hand limit and right-hand limit are equal, but \( f(a) \) is undefined?

**Tip**: Always check both the function's value and the limit when determining continuity at a specific point.

Explain why the function is discontinuous at the given number a. f(x) = 1/(x + 5), a = -5. f(-5) and lim x→-5 f(x) are finite, but are not equal. f(-5) is undefined. lim x→-5+ f(x) and lim x→-5− f(x) are not finite, and are not equal. lim x→-5 f(x) does not exist. none of the above.

Explore why the function f(x) = 1/(x + 5) is discontinuous at a = -5. Learn about limits, the behavior of rational functions near undefined points, and conditions for discontinuity. Suitable for high school students in Grades 11-12.

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