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The image contains a mathematical problem where the function \( g(x) = \frac{x^2 + x - 30}{|x - 5|} \) is given, and the task is to evaluate limits and determine whether the derivative of the function exists at certain points. Here's how we can approach it:

### Problem (a):
Evaluate the following limits:
1. \( \lim_{x \to 5^+} g(x) \)
2. \( \lim_{x \to 5^-} g(x) \)

### Step-by-Step Solution:
1. **Factor the numerator:**
   \[
   x^2 + x - 30 = (x - 5)(x + 6)
   \]
   Thus, the function becomes:
   \[
   g(x) = \frac{(x - 5)(x + 6)}{|x - 5|}
   \]

2. **Simplify the function based on the sign of \( x - 5 \):**
   - For \( x > 5 \), \( |x - 5| = x - 5 \), so:
     \[
     g(x) = \frac{(x - 5)(x + 6)}{x - 5} = x + 6
     \]
   - For \( x < 5 \), \( |x - 5| = -(x - 5) \), so:
     \[
     g(x) = \frac{(x - 5)(x + 6)}{-(x - 5)} = -(x + 6)
     \]

3. **Evaluate the limits:**
   - As \( x \to 5^+ \), \( g(x) = x + 6 \), so:
     \[
     \lim_{x \to 5^+} g(x) = 5 + 6 = 11
     \]
   - As \( x \to 5^- \), \( g(x) = -(x + 6) \), so:
     \[
     \lim_{x \to 5^-} g(x) = -(5 + 6) = -11
     \]

### Conclusion for Part (a):
- \( \lim_{x \to 5^+} g(x) = 11 \)
- \( \lim_{x \to 5^-} g(x) = -11 \)

### Problem (b):
Does \( \lim_{x \to 5} g(x) \) exist?

Since the left-hand and right-hand limits are not equal (\( 11 \neq -11 \)), the two-sided limit does **not exist** at \( x = 5 \).

Would you like further details on these steps or additional questions? Here are some related questions to deepen understanding:

1. What is the definition of a one-sided limit?
2. Why does the behavior of the absolute value function change depending on whether \( x > 5 \) or \( x < 5 \)?
3. How do you determine if a limit exists at a given point?
4. What is the difference between a continuous function and a discontinuous one?
5. How can factoring help in simplifying limit problems?

**Tip**: When solving limit problems involving absolute values, always consider splitting the function based on different intervals of \( x \) where the absolute value expression changes.

Let g(x) = (x^2 + x - 30) / |x - 5|. (a) Find the following limits: (i) lim (x→5+) g(x), (ii) lim (x→5-) g(x), (b) Does lim g(x) (x→5) exist?

Explore how to solve a limit problem involving an absolute value function. Learn how to factor and evaluate one-sided limits for the function g(x) = (x^2 + x - 30) / |x - 5|. Suitable for high school students in Grades 10-12.

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