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.