To solve the problem, we need to analyze the given conditions and match them with the choices provided.

1. Condition Analysis:

• The function $g(x)$ is increasing for $x < 1$ and $x > 1$.
• The limit $\lim_{x \to 1} g(x) = 5$.

2. Analysis of the Given Function $g(x)$:

• $g(x) = \frac{2x^2 + 3x - 4}{x - 1}$ for $x \neq 1$.
• $g(x) = 5$ for $x = 1$.

Check the limit as $x$ approaches 1:

• The expression $\frac{2x^2 + 3x - 4}{x - 1}$ suggests there might be a factor that cancels out the denominator at $x = 1$. Let's find this factor:

  $2x^2 + 3x - 4$

  • Factor the numerator: $2x^2 + 3x - 4 = (2x - 1)(x + 4)$
  • The function can be rewritten as: $g(x) = \frac{(2x - 1)(x + 4)}{x - 1}$
  • As $x$ approaches 1, the function does not simplify easily since the denominator does not cancel out with the numerator. However, if we evaluate the limit: $\lim_{x \to 1} g(x) = \lim_{x \to 1} \frac{2x^2 + 3x - 4}{x - 1}$
  • As $x$ approaches 1, the numerator goes to 1 and the denominator goes to 0, which results in an undefined situation.

  However, since we are given that the limit $\lim_{x \to 1} g(x) = 5$, this implies there is a removable discontinuity and the limit might simplify to 5 when the undefined portion is considered carefully or approached by limits, but the exact cancellation is not apparent.

3. Table Analysis (II):

• Looking at the table values:
  • As $x$ approaches 1 from both sides (0.999, 1.001), $g(x)$ approaches very large values (5000 and -5000), indicating a significant jump or break at $x = 1$.
  • This suggests that the function values are not continuous at $x = 1$, which could align with the function definition and limit condition given.

4. Graph Analysis (III):

• The graph shows a break at $x = 1$ and a continuous line elsewhere, which matches with a possible piecewise definition that jumps or has a significant change at $x = 1$.

Conclusion:

• Based on the condition $g(x)$ is increasing both for $x < 1$ and $x > 1$, the function's behavior around $x = 1$, and the graph, we can conclude:
• Graph III is consistent with the given function $g(x)$.
• The table (II) also suggests a discontinuity or significant change at $x = 1$.

Answer:

• The correct answer is D) I and III only.

Would you like a detailed explanation on any specific part, or do you have any other questions?

Relative Questions:

1. How does factoring the numerator help in understanding the function's limit behavior?
2. What are the implications of having a removable discontinuity in a function?
3. Why is it important to check both sides of the limit when evaluating discontinuities?
4. How can graph analysis help in understanding the behavior of a function near critical points?
5. What does it mean for a function to be "increasing" and how can this be determined from a graph?

Tip:

Always verify limit behavior by evaluating the function from both sides of the point of interest to confirm any discontinuity or removable discontinuity.