We are asked to evaluate the limit:

$\lim_{x \to 2} \left( -\frac{3(x - 2)}{4 |x - 2|} \right)$

Step 1: Break it into two cases

Since the expression involves $|x - 2|$, we need to handle this piecewise, depending on whether $x$ is approaching 2 from the left (i.e., $x \to 2^-$) or from the right (i.e., $x \to 2^+$).

Case 1: $x \to 2^+$ (Approaching from the right)

When $x > 2$, we have $|x - 2| = x - 2$. Therefore, the expression becomes:

$\lim_{x \to 2^+} \left( -\frac{3(x - 2)}{4(x - 2)} \right)$

Simplifying this:

$\lim_{x \to 2^+} \left( -\frac{3}{4} \right) = -\frac{3}{4}$

Case 2: $x \to 2^-$ (Approaching from the left)

When $x < 2$, we have $|x - 2| = -(x - 2)$. Therefore, the expression becomes:

$\lim_{x \to 2^-} \left( -\frac{3(x - 2)}{4(-(x - 2))} \right)$

Simplifying this:

$\lim_{x \to 2^-} \left( \frac{3}{4} \right) = \frac{3}{4}$

Step 2: Conclusion

We observe that the limit from the right is $-\frac{3}{4}$ and the limit from the left is $\frac{3}{4}$. Since these two one-sided limits are not equal, the two-sided limit does not exist.

Thus, the limit is:

$\lim_{x \to 2} \left( -\frac{3(x - 2)}{4 |x - 2|} \right) \text{ does not exist.}$

Would you like further details or have any questions?

Here are some related questions:

1. How does the absolute value function impact limits?
2. What are one-sided limits and why are they important?
3. Can a function have different one-sided limits?
4. How do you evaluate limits involving absolute value?
5. What is the significance of piecewise functions in limit evaluation?

Tip: When working with absolute values in limits, always consider both sides (left and right) to ensure the correct limit behavior.