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Let's analyze and solve the given limits step-by-step.

### Problem #2: Evaluate the following limits

#### (a) \(\lim\limits_{h \to 0} \frac{\sqrt{67 - 3(x+h)} - \sqrt{67 - 3x}}{h}\)

1. **Rationalize the numerator**: Multiply both numerator and denominator by the conjugate, \(\sqrt{67 - 3(x+h)} + \sqrt{67 - 3x}\), to eliminate the square roots.

   \[
   \lim_{h \to 0} \frac{ \left(\sqrt{67 - 3(x+h)} - \sqrt{67 - 3x} \right) \left( \sqrt{67 - 3(x+h)} + \sqrt{67 - 3x} \right) }{ h \left( \sqrt{67 - 3(x+h)} + \sqrt{67 - 3x} \right)}
   \]

   Simplifying the numerator using the identity \( (a - b)(a + b) = a^2 - b^2 \):

   \[
   = \lim_{h \to 0} \frac{ (67 - 3(x+h)) - (67 - 3x) }{ h \left( \sqrt{67 - 3(x+h)} + \sqrt{67 - 3x} \right) }
   \]

   This reduces to:

   \[
   \lim_{h \to 0} \frac{ -3h }{ h \left( \sqrt{67 - 3(x+h)} + \sqrt{67 - 3x} \right) }
   \]

2. **Simplify**: Cancel \( h \) in the numerator and denominator (assuming \( h \neq 0 \)):

   \[
   = \lim_{h \to 0} \frac{-3}{ \sqrt{67 - 3(x+h)} + \sqrt{67 - 3x}}
   \]

3. **Substitute \( h = 0 \)**:

   \[
   = \frac{-3}{2\sqrt{67 - 3x}}
   \]

So, the limit is:

\[
\lim_{h \to 0} \frac{\sqrt{67 - 3(x+h)} - \sqrt{67 - 3x}}{h} = \frac{-3}{2\sqrt{67 - 3x}}
\]

#### (b) \(\lim\limits_{x \to 0^-} \frac{x^2 - 7x}{x}\)

Simplify the expression:

\[
\frac{x^2 - 7x}{x} = x - 7
\]

Now, take the limit as \( x \to 0^- \):

\[
\lim_{x \to 0^-} (x - 7) = -7
\]

So, the limit is:

\[
\lim_{x \to 0^-} \frac{x^2 - 7x}{x} = -7
\]

#### (c) \(\lim\limits_{x \to 0^+} \frac{x^2 - 4x}{x}\)

Simplify the expression:

\[
\frac{x^2 - 4x}{x} = x - 4
\]

Now, take the limit as \( x \to 0^+ \):

\[
\lim_{x \to 0^+} (x - 4) = -4
\]

So, the limit is:

\[
\lim_{x \to 0^+} \frac{x^2 - 4x}{x} = -4
\]

---

Would you like a more detailed breakdown of any specific part? Here are five questions related to this problem:

1. Why is rationalizing the numerator helpful in evaluating the limit in part (a)?
2. What happens if you substitute \( h = 0 \) directly in part (a)?
3. Can you explain why the expression simplifies to \( x - 7 \) in part (b)?
4. How does the concept of one-sided limits apply in parts (b) and (c)?
5. What is the significance of approaching \( x \) from the left and right separately?

**Tip**: Always simplify the expression before taking the limit to see if you can resolve indeterminate forms.

Problem #2: Evaluate the following limits 
(a) lim_{h -> 0} [(sqrt(67 - 3(x + h)) - sqrt(67 - 3x)) / h] 
(b) lim_{x -> 0^-} [(x^2 - 7x) / x] 
(c) lim_{x -> 0^+} [(x^2 - 4x) / x]

Learn how to evaluate limits involving rationalization techniques and one-sided limits. This problem covers key calculus concepts such as rationalization of square roots, limits at x -> 0, and one-sided limits. Suitable for students in advanced high school or college-level calculus.

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