To determine the type of discontinuity at $x = 2$ for each of the given functions, we need to analyze their behavior at that point.

Part (a): $f(x) = \frac{x - 2}{|x - 2|}$

1. Definition Check: Notice that the function involves an absolute value in the denominator, which will affect the behavior of $f(x)$ differently based on whether $x$ is greater than or less than 2.

2. Behavior as $x \to 2^+$:

   • When $x > 2$, $|x - 2| = x - 2$.
   • Substituting into $f(x)$: $f(x) = \frac{x - 2}{x - 2} = 1$

3. Behavior as $x \to 2^-$:

   • When $x < 2$, $|x - 2| = -(x - 2)$.
   • Substituting into $f(x)$: $f(x) = \frac{x - 2}{-(x - 2)} = -1$

4. Conclusion:

   • As $x \to 2$ from the left, $f(x) \to -1$.
   • As $x \to 2$ from the right, $f(x) \to 1$.
   • Since the left-hand limit and the right-hand limit are not equal, there is a jump discontinuity at $x = 2$ for $f(x)$.

Part (b): $g(x) = \frac{\sqrt{x} - \sqrt{2}}{x - 2}$

1. Direct Substitution: Plugging in $x = 2$ directly gives an indeterminate form $\frac{0}{0}$, so we’ll use L'Hôpital's Rule.

2. Applying L'Hôpital's Rule:

   • Differentiate the numerator and the denominator with respect to $x$.
     • The derivative of $\sqrt{x} - \sqrt{2}$ with respect to $x$ is $\frac{1}{2\sqrt{x}}$.
     • The derivative of $x - 2$ with respect to $x$ is 1.
   • Applying L'Hôpital's Rule: $\lim_{x \to 2} g(x) = \lim_{x \to 2} \frac{\frac{1}{2\sqrt{x}}}{1} = \frac{1}{2\sqrt{2}}$

3. Conclusion:

   • The limit exists and is finite, so $g(x)$ has a removable discontinuity at $x = 2$.
   • We could redefine $g(x)$ at $x = 2$ as $\frac{1}{2\sqrt{2}}$ to make it continuous at that point.

Would you like a deeper breakdown of these steps or have any questions?

Here are five related questions to further explore this topic:

1. How do different types of discontinuities (jump, removable, infinite) affect the continuity of a function?
2. Can all discontinuities be removed by redefining the function at a point?
3. How does the absolute value function impact the limits of functions as $x$ approaches a given point?
4. When is it appropriate to use L'Hôpital's Rule for evaluating limits?
5. How can piecewise functions be used to define functions with jump discontinuities?

Tip: For functions with absolute values, consider the behavior on either side of the point in question, as it often leads to different expressions for the limit.