To find the limit of $G(x) = |2x - 3| - 4$ as $x$ approaches $\frac{3}{2}$, we can follow these steps:

1. Substitute $x = \frac{3}{2}$ directly into the expression:

   $G\left(\frac{3}{2}\right) = \left|2\left(\frac{3}{2}\right) - 3\right| - 4$

2. Simplify inside the absolute value:

   $G\left(\frac{3}{2}\right) = \left|3 - 3\right| - 4 = |0| - 4 = 0 - 4 = -4$

3. Evaluate the limit:

   Since the function $G(x) = |2x - 3| - 4$ is continuous at $x = \frac{3}{2}$, the limit as $x$ approaches $\frac{3}{2}$ is simply the value of the function at that point.

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

So, the limit is:

$\lim_{{x \to \frac{3}{2}}} G(x) = -4$

Would you like more details, or do you have any other questions?

Here are some related questions to further explore this topic:

1. What is the derivative of $G(x) = |2x - 3| - 4$ with respect to $x$?
2. How would the limit change if $G(x) = |2x - 3| - 4$ was evaluated as $x$ approaches a different value?
3. Can you explain the concept of continuity and how it relates to evaluating limits?
4. How would you graph $G(x) = |2x - 3| - 4$ and interpret the result?
5. What happens to the limit of $G(x) = |2x - 3| - 4$ as $x$ approaches infinity?

Tip: When evaluating limits involving absolute values, consider the points where the expression inside the absolute value changes sign, as these are critical points to analyze the behavior of the function.