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.