Evaluate: `lim_(x rarr 3) (|x-3|)/(x-3)`.

Answer

To evaluate the limit \( \lim_{x \to 3} \frac{|x-3|}{x-3} \), we will analyze the behavior of the function as \( x \) approaches 3 from both the left and the right.

### Step 1: Analyze the left-hand limit
We start by finding the left-hand limit as \( x \) approaches 3 from the left (denoted as \( x \to 3^- \)):
\[ \lim_{x \to 3^-} \frac{|x-3|}{x-3} \]
For \( x < 3 \), \( |x-3| = -(x-3) \). Therefore, we can rewrite the limit:
\[ \lim_{x \to 3^-} \frac{-(x-3)}{x-3} = \lim_{x \to 3^-} -1 = -1 \]

### Step 2: Analyze the right-hand limit
Next, we find the right-hand limit as \( x \) approaches 3 from the right (denoted as \( x \to 3^+ \)):
\[ \lim_{x \to 3^+} \frac{|x-3|}{x-3} \]
For \( x > 3 \), \( |x-3| = x-3 \). Therefore, we can rewrite the limit:
\[ \lim_{x \to 3^+} \frac{x-3}{x-3} = \lim_{x \to 3^+} 1 = 1 \]

### Step 3: Compare the left-hand and right-hand limits
Now we compare the two limits we found:
- Left-hand limit: \( -1 \)
- Right-hand limit: \( 1 \)

Since the left-hand limit and the right-hand limit are not equal, we conclude that the overall limit does not exist.

### Final Answer
\[ \lim_{x \to 3} \frac{|x-3|}{x-3} \text{ does not exist.} \]