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.