lim G(x), given G(x) = |2x - 3| - 4 as x approaches to 3/2

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.

Explore the limit of the function G(x) = |2x - 3| - 4 as x approaches 3/2. Learn how to evaluate limits involving absolute value functions with a step-by-step solution. Suitable for advanced high school students and anyone interested in calculus concepts.

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