The problem presented involves evaluating limits and determining if the limit is $\infty$, $-\infty$, or if the limit does not exist. Let's break down the questions one by one:

Given:

$\lim_{x \to -5^{+}} \frac{(x - 1)(x - 5)}{x + 5}$

(a) $\lim_{x \to -5^{+}} \frac{(x - 1)(x - 5)}{x + 5}$

Step 1: Check the denominator at $x = -5$:

As $x \to -5^{+}$, the denominator $(x + 5) \to 0^{+}$ because we are approaching from the right-hand side (values of $x$ greater than -5).

Step 2: Behavior of the numerator:

The numerator $(x - 1)(x - 5)$ at $x = -5^{+}$ gives: $(x - 1)(x - 5) = (-5 - 1)(-5 - 5) = (-6)(-10) = 60$ Thus, the numerator remains 60.

Step 3: Conclusion:

As $x \to -5^{+}$, the numerator is positive and the denominator approaches 0 from the positive side, so the fraction approaches $+\infty$.

Answer: $+\infty$.

(b) $\lim_{x \to -5^{-}} \frac{(x - 1)(x - 5)}{x + 5}$

Step 1: Check the denominator at $x = -5$:

As $x \to -5^{-}$, the denominator $(x + 5) \to 0^{-}$, since we are approaching from the left-hand side (values of $x$ less than -5).

Step 2: Behavior of the numerator:

The numerator remains the same as above, i.e., $(x - 1)(x - 5) = 60$.

Step 3: Conclusion:

As $x \to -5^{-}$, the numerator is positive (60), and the denominator approaches 0 from the negative side. Thus, the fraction approaches $-\infty$.

Answer: $-\infty$.

(c) $\lim_{x \to -5} \frac{(x - 1)(x - 5)}{x + 5}$

In this case, we are considering the two-sided limit as $x \to -5$.

Step 1: Analyze the one-sided limits:

From parts (a) and (b), we know that as $x \to -5^{+}$, the limit approaches $+\infty$, and as $x \to -5^{-}$, the limit approaches $-\infty$.

Step 2: Conclusion:

Since the one-sided limits are not equal, the two-sided limit does not exist.

Answer: The limit does not exist.

Let me know if you need more details on any step!

5 Related Questions:

1. What is the difference between one-sided and two-sided limits?
2. How do you determine whether a limit goes to infinity or negative infinity?
3. What happens to the behavior of a rational function as its denominator approaches zero?
4. Can a limit exist if one-sided limits approach different infinities?
5. How can factoring help simplify limits of rational functions?

Tip:

Always check the behavior of both the numerator and denominator independently when evaluating limits that involve rational functions. This helps in determining the direction of the limit (whether it approaches $\infty$, $-\infty$, or does not exist).