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.