The problem asks to evaluate the limit: $ \lim_{x \to 3^+} \frac{x^2 - 3x}{|x-3|} $

1. Problem Description

The problem asks to evaluate the limit:

$\lim_{x \to 3^+} \frac{x^2 - 3x}{|x-3|}$

2. Solution Steps

First, we can factor the numerator:

$x^2 - 3x = x(x-3)$

Then the expression becomes:

$\lim_{x \to 3^+} \frac{x(x-3)}{|x-3|}$

Since $x \to 3^+$, $x$ is approaching 3 from values greater than

3. This means that $x-3 > 0$, and hence $|x-3| = x-3$.

So the expression becomes:

$\lim_{x \to 3^+} \frac{x(x-3)}{x-3}$

We can cancel out the $(x-3)$ terms as $x$ approaches 3 but is not equal to 3:

$\lim_{x \to 3^+} x$

Now we can evaluate the limit by plugging in $x=3$:

$\lim_{x \to 3^+} x = 3$

3. Final Answer

The final answer is 3.