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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).

Determine the following limits using ∞ or -∞ when appropriate, or state that they do not exist: (a) lim (x->-5+) (x-1)(x-5)/(x+5), (b) lim (x->-5-) (x-1)(x-5)/(x+5), (c) lim (x->-5) (x-1)(x-5)/(x+5).

Learn how to evaluate one-sided and two-sided limits for the rational function (x-1)(x-5)/(x+5) as x approaches -5. This problem covers key calculus concepts including infinity, negative infinity, and determining whether a limit exists.

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