To find the limit of the function \( G(x) = |2x - 3| - 4 \) as \( x \) approaches a specific value, we need to consider the behavior of the absolute value function \( |2x - 3| \) as \( x \) approaches that value.

Since you haven't specified the point at which you're taking the limit, let's examine the limit as \( x \) approaches two key points: \( x = \frac{3}{2} \) and general limits for \( x \to \infty \) and \( x \to -\infty \).

### Case 1: Limit as \( x \to \frac{3}{2} \)
At \( x = \frac{3}{2} \), the expression inside the absolute value becomes zero:

\[
2x - 3 = 2\left(\frac{3}{2}\right) - 3 = 3 - 3 = 0
\]

So, at \( x = \frac{3}{2} \), \( G(x) = |0| - 4 = -4 \).

Thus,

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

### Case 2: Limit as \( x \to \infty \)
As \( x \) becomes very large, the expression \( 2x - 3 \) is positive, and the absolute value does not affect the sign:

\[
|2x - 3| = 2x - 3
\]

So,

\[
G(x) = 2x - 3 - 4 = 2x - 7
\]

As \( x \to \infty \):

\[
\lim_{x \to \infty} G(x) = \lim_{x \to \infty} (2x - 7) = \infty
\]

### Case 3: Limit as \( x \to -\infty \)
As \( x \) becomes very large in the negative direction, \( 2x - 3 \) is negative, so:

\[
|2x - 3| = -(2x - 3) = -2x + 3
\]

Thus,

\[
G(x) = -2x + 3 - 4 = -2x - 1
\]

As \( x \to -\infty \):

\[
\lim_{x \to -\infty} G(x) = \lim_{x \to -\infty} (-2x - 1) = \infty
\]

### Summary
- \(\lim_{x \to \frac{3}{2}} G(x) = -4\)
- \(\lim_{x \to \infty} G(x) = \infty\)
- \(\lim_{x \to -\infty} G(x) = \infty\)

Would you like more details on any of these steps or have any questions?

### Follow-Up Questions:
1. What are the general properties of absolute value functions?
2. How do you determine the left-hand and right-hand limits for a piecewise function?
3. How does the behavior of functions differ as \( x \to \infty \) versus \( x \to -\infty \)?
4. Can you explain how to graph \( G(x) = |2x - 3| - 4 \) step by step?
5. How do discontinuities affect the limits of functions?

**Tip:** When dealing with absolute value functions, always consider breaking the problem into cases based on the point where the expression inside the absolute value equals zero.

Explore the limit of the absolute value function G(x) = |2x - 3| - 4 as x approaches specific values. Learn detailed cases for limits as x approaches a constant, infinity, and negative infinity. Suitable for advanced high school students and calculus enthusiasts.

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