Decide from the graph whether each limit exists. If a limit​ exists, estimate its value.
​(a) lim f(x) x-> 7                                                        
​(b) lim f(x) x->6

To determine whether a limit exists and estimate its value, you need to inspect the behavior of the function \( f(x) \) as \( x \) approaches specific values from both sides on the graph. Let's assume you are referring to a piecewise or continuous function, as these are typical in limit problems. Here's how to approach it:

### (a) \( \lim_{{x \to 7}} f(x) \)
- Look at the value of \( f(x) \) as \( x \) approaches 7 from both the left-hand side (\( x \to 7^- \)) and the right-hand side (\( x \to 7^+ \)).
- If the graph approaches the same value from both directions, then the limit exists.
- **Does it exist?**
    - If the left-hand and right-hand values are the same, the limit exists.
    - **Estimate**: If the function approaches a certain value (say \( L \)) near \( x = 7 \), then the limit is \( L \).

### (b) \( \lim_{{x \to 6}} f(x) \)
- Again, examine the graph at \( x = 6 \) from both sides. 
- **Does it exist?**
    - Check if the left-hand and right-hand limits are equal. If they are, the limit exists.
    - **Estimate**: If the graph approaches a specific value near \( x = 6 \), the limit can be estimated as that value.

### Conclusion
- If the graph shows a discontinuity (jump, vertical asymptote, or oscillation) near \( x = 7 \) or \( x = 6 \), the limit may not exist.
- If the function smoothly approaches the same value from both sides, the limit exists and can be estimated from the graph.

If you can upload a graph, I can give a more detailed answer. Would you like me to explain more about how limits behave around discontinuities or provide further clarification?

#### 5 Related Questions:
1. What happens if the left-hand and right-hand limits do not match?
2. Can a function have a limit at a point where it is not defined?
3. What are the types of discontinuities that affect limits?
4. How can you estimate a limit if the graph approaches infinity?
5. How does the concept of one-sided limits help in limit problems?

#### Tip:
Always check both sides of the graph (left and right) when estimating limits to ensure they match for the limit to exist.

Decide from the graph whether each limit exists. If a limit​ exists, estimate its value. (a) lim f(x) x->7 (b) lim f(x) x->6

Understand how to decide whether limits exist at x = 6 and x = 7 by examining the behavior of the function f(x) on a graph. This explanation covers concepts of left-hand and right-hand limits, continuity, and the conditions for limit existence. Suitable for Grades 10-12.

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