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Let's analyze and solve the limits step-by-step from the provided image.

### 2. \(\lim_{x \to 6^+} \frac{|2 - |2 - x||}{2 - |x - 2|}\)

We need to handle the absolute values carefully based on the behavior of \(x\) as it approaches 6 from the right (\(x \to 6^+\)).

#### Absolute Value Analysis:
- \( |2 - x| \): Since \(x \to 6^+\), \(x > 6\), so \( |2 - x| = x - 2 \).
- \( |2 - (2 - x)| = |x|\): This simplifies to \(x - 2\).

So, the expression becomes:
\[
\frac{|2 - (x - 2)|}{2 - (x - 2)} = \frac{0}{x-2} \implies 0
\]

### 3. \(\lim_{x \to 2} \left( \frac{1}{3 - x^2 + 5} \left( 1 - \frac{1}{1 - \sqrt{x - 1}} \right) \right)\)

First, simplify the terms as \(x \to 2\):

- For \( \frac{1}{3 - x^2 + 5} \), substitute \(x = 2\):
\[
3 - (2)^2 + 5 = 3 - 4 + 5 = 4.
\]
So the fraction becomes \( \frac{1}{4} \).

- Now for the second part: \( 1 - \frac{1}{1 - \sqrt{x - 1}} \).
As \(x \to 2\), \( \sqrt{x - 1} = \sqrt{2 - 1} = 1 \), so the expression inside becomes:
\[
1 - \frac{1}{1 - 1} \text{ is undefined because of division by 0}.
\]

Thus, the limit does not exist due to a singularity at \(x = 2\).

### 4. \(\lim_{x \to 3} \frac{2 - x}{3^x + 3^{3 - x} - 28}\)

Let’s first check if direct substitution works.

- Substitute \(x = 3\) into the numerator: \(2 - 3 = -1\).
- Now into the denominator:
  \[
  3^3 + 3^{3 - 3} - 28 = 27 + 1 - 28 = 0.
  \]
  
So we have the indeterminate form \(\frac{-1}{0}\), which tends to \(-\infty\). Therefore, the limit does not exist and goes to \(-\infty\).

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Would you like more details on these solutions or explanations for the methods? Here are five related questions that might help deepen your understanding:

1. What is the significance of calculating limits from the right-hand side (\(x \to 6^+\))?
2. How do we deal with absolute value expressions in limits?
3. Why does the second limit lead to a singularity or undefined behavior?
4. How do we analyze limits that tend towards infinity or negative infinity?
5. What are common techniques to simplify complex limit expressions?

**Tip:** When evaluating limits involving absolute values, always consider the direction from which \(x\) is approaching, as this affects how the absolute value expressions behave.

Solve the following limits: 2. lim_{x -> 6^+} |2 - |2 - x|| / (2 - |x - 2|), 3. lim_{x -> 2} (1 / (3 - x^2 + 5)) (1 - (1 / (1 - sqrt(x - 1)))), 4. lim_{x -> 3} (2 - x) / (3^x + 3^(3 - x) - 28).

This page explores how to solve complex limits involving absolute values and singularities. Topics covered include limits approaching from the right, handling indeterminate forms, and singularities. Ideal for college-level calculus students.

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