 Does 

ModifyingBelow lim With x right arrow 3 f left parenthesis x right parenthesislimx→3f(x) 

​exist? If​ so, what is​ it? If​ not, why​ not? Select the correct choice below​ and, if​ necessary, fill in any answer box in your choice.

To determine whether \(\lim_{{x \to 3}} f(x)\) exists, we need to compare the left-hand limit \(\lim_{{x \to 3^-}} f(x)\) and the right-hand limit \(\lim_{{x \to 3^+}} f(x)\).

### Recap of the Limits:
- **Left-hand limit:** \(\lim_{{x \to 3^-}} f(x) = 3\)
- **Right-hand limit:** \(\lim_{{x \to 3^+}} f(x) = 5\)

### Conclusion:
The overall limit \(\lim_{{x \to 3}} f(x)\) exists only if both the left-hand limit and the right-hand limit are equal. In this case:

\[
\lim_{{x \to 3^-}} f(x) = 3 \quad \text{and} \quad \lim_{{x \to 3^+}} f(x) = 5
\]

Since the left-hand limit (3) and the right-hand limit (5) are not equal, the overall limit \(\lim_{{x \to 3}} f(x)\) **does not exist**.

### Answer:
The correct choice is:

- **The limit does not exist** because the left-hand limit and the right-hand limit as \(x\) approaches 3 are not equal.

Would you like to go through the next part of the problem or have any other questions?

Here are five related questions that might interest you:

1. What are some examples of functions where the limit at a point does not exist?
2. How can you graphically identify if the limit at a point does not exist?
3. What role do discontinuities play in determining whether a limit exists?
4. How would you define a limit that does exist for a continuous function?
5. Can a function have a limit at a point even if it is not defined at that point?

**Tip:** For a limit to exist at a particular point, the function must approach the same value from both the left and the right sides as \(x\) approaches that point.

Learn about the existence of the limit lim[x→3] f(x) in calculus. Understand how left-hand and right-hand limits determine the overall limit's existence. This problem demonstrates the concept of limits and their application in analyzing functions. Suitable for advanced high school students.

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