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.