The problem asks to evaluate the following limits for the function $f(x) = |x - 2|$.

1. First Limit: $\lim_{x \to 2^-} \frac{f(x) - f(2)}{x - 2}$ Here, $f(x) = |x - 2|$ and $f(2) = |2 - 2| = 0$, so the expression becomes: $\frac{|x - 2| - 0}{x - 2} = \frac{|x - 2|}{x - 2}$ When $x \to 2^-$ (approaching from the left), $x - 2$ is negative, so $|x - 2| = -(x - 2)$. Thus, the limit becomes: $\lim_{x \to 2^-} \frac{-(x - 2)}{x - 2} = -1$

2. Second Limit: $\lim_{x \to 2^+} \frac{f(x) - f(2)}{x - 2}$ Again, using $f(2) = 0$, the expression simplifies to: $\frac{|x - 2|}{x - 2}$ When $x \to 2^+$ (approaching from the right), $x - 2$ is positive, so $|x - 2| = x - 2$. The limit becomes: $\lim_{x \to 2^+} \frac{x - 2}{x - 2} = 1$

Final Answers:

1. $\lim_{x \to 2^-} \frac{f(x) - f(2)}{x - 2} = -1$
2. $\lim_{x \to 2^+} \frac{f(x) - f(2)}{x - 2} = 1$

Since the two one-sided limits do not agree, the function $f(x) = |x - 2|$ is not differentiable at $x = 2$.

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Related Questions:

1. How do you formally define differentiability at a point?
2. What are the conditions for a piecewise function to be differentiable at a point?
3. Can a function be continuous but not differentiable? Provide examples.
4. What is the geometric interpretation of differentiability at a point?
5. How can we compute limits involving absolute value functions for different points?

Tip: When evaluating limits for absolute value functions, always consider the behavior of the function from both the left and right to handle cases where the absolute value expression changes form.