Use the graph below to find the integer value(s) of x where the limit does not equal a finite value as x approaches those integer value(s).

 A graph is shown beginning at the open point negative two comma negative four continuing to the open point negative one negative one up to a maximum at zero comma zero and back down to the open point one comma negative one.  The graph begins again at the closed point one comma two and then condinues down to infinity along the asymptote x equals three then from negative infinity along the asymptote of x equals three the graph increases  to the closed point five comma zero.  A noncontinuous closed point exists at negative one comma negative two.

To find the integer values of \(x\) where the limit does not equal a finite value as \(x\) approaches those values, let's analyze the described graph step-by-step.

1. **Discontinuities due to open points:**
   - The graph has open points at \( (-2, -4) \), \( (-1, -1) \), and \( (1, -1) \). However, open points alone do not necessarily indicate that the limit does not exist. We need to consider other factors such as jumps or vertical asymptotes.

2. **Non-continuous closed point:**
   - There is a **closed point** at \( (-1, -2) \), but the graph reaches the open point \( (-1, -1) \). This suggests a **jump discontinuity** at \(x = -1\), because the function jumps from \( (-1, -1) \) to \( (-1, -2) \). At jump discontinuities, the left-hand and right-hand limits differ, meaning the overall limit does not exist at \(x = -1\).

3. **Asymptote at \(x = 3\):**
   - The graph approaches **vertical asymptotes** as \(x\) approaches 3 from both the left and the right, going toward \( \infty \) and \( -\infty \), respectively. This implies that the limit does not exist at \(x = 3\), since the function does not approach a finite value.

### Conclusion:

The integer values of \(x\) where the limit does not equal a finite value are:
- \(x = -1\) (due to the jump discontinuity)
- \(x = 3\) (due to the vertical asymptote)

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Would you like more details about this process or further clarification? Here are five follow-up questions you might find interesting:

1. What are the different types of discontinuities in a graph?
2. How do we determine if a limit exists at a point of discontinuity?
3. How do asymptotes affect the existence of limits?
4. Can a function have a limit at an open point?
5. How do we interpret left-hand and right-hand limits on a graph?

**Tip:** When analyzing graphs for limits, always check for discontinuities, jumps, holes, and vertical asymptotes, as these often indicate where limits do not exist.

Use the graph below to find the integer value(s) of x where the limit does not equal a finite value as x approaches those integer value(s). A graph is shown beginning at the open point negative two comma negative four continuing to the open point negative one negative one up to a maximum at zero comma zero and back down to the open point one comma negative one. The graph begins again at the closed point one comma two and then continues down to infinity along the asymptote x equals three then from negative infinity along the asymptote of x equals three the graph increases to the closed point five comma zero. A noncontinuous closed point exists at negative one comma negative two.

Analyze a graph with jump discontinuities and vertical asymptotes to find the integer values of x where the limit does not exist. Explore key mathematical concepts like limits, discontinuities, and asymptotes, ideal for Grades 11-12.

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