Let's evaluate each statement based on the function provided:

Given function:

$f(x) = \frac{x-3}{|x-3|}, \text{ for } x \neq 3,$ $f(3) = 3.$

1. Evaluating $\lim_{x \to 3^+} f(x)$:

• For $x > 3$, $x - 3 > 0$, so $|x - 3| = x - 3$.
• Therefore, when $x > 3$, $f(x) = \frac{x - 3}{x - 3} = 1.$
• Hence, $\lim_{x \to 3^+} f(x) = 1$.

2. Evaluating $\lim_{x \to 3^-} f(x)$:

• For $x < 3$, $x - 3 < 0$, so $|x - 3| = -(x - 3)$.
• Therefore, when $x < 3$, $f(x) = \frac{x - 3}{-(x - 3)} = -1.$
• Hence, $\lim_{x \to 3^-} f(x) = -1$.

3. Evaluating $\lim_{x \to 3} f(x)$:

• Since the left-hand limit is $-1$ and the right-hand limit is $1$, the overall limit as $x \to 3$ does not exist.
• Thus, $\lim_{x \to 3} f(x) \neq 3$ and in fact, doesn't exist at all.

Conclusion:

• Statement A (Limit as x approaches 3 from the right $f(x) = 1$) is true.
• Statement B (Limit as x approaches 3 from the left $f(x) = -1$) is true.
• Statement C (Limit as x approaches 3 $f(x) = 3$) is false.

Would you like any more details or have further questions?

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Here are 5 related questions you might find useful:

1. What is the definition of a one-sided limit?
2. How do we check if a function is continuous at a point?
3. How do absolute value functions affect limits?
4. What are the conditions for a limit to exist at a point?
5. Can a limit exist if the left-hand and right-hand limits are different?

Tip: Always check the left-hand and right-hand limits separately when dealing with absolute value functions or piecewise functions.