Evaluate the following limits :
`Lim_( x to 5^(+)) (x-5)/(|x-5|)`

Answer

To evaluate the limit

\[ \lim_{x \to 5^+} \frac{x - 5}{|x - 5|}, \]

we will follow these steps:

### Step 1: Understand the behavior of \( |x - 5| \) as \( x \) approaches \( 5 \) from the right.
When \( x \) approaches \( 5 \) from the right (denoted as \( 5^+ \)), \( x \) is greater than \( 5 \). Therefore, \( x - 5 \) is positive.

### Step 2: Simplify the expression.
Since \( x - 5 \) is positive when \( x \to 5^+ \), we can write:

\[ |x - 5| = x - 5. \]

Thus, the limit can be rewritten as:

\[ \lim_{x \to 5^+} \frac{x - 5}{|x - 5|} = \lim_{x \to 5^+} \frac{x - 5}{x - 5}. \]

### Step 3: Cancel the terms.
Now, we can simplify the expression:

\[ \frac{x - 5}{x - 5} = 1 \quad \text{(for } x \neq 5\text{)}. \]

### Step 4: Evaluate the limit.
Now, we take the limit:

\[ \lim_{x \to 5^+} 1 = 1. \]

### Final Answer:
Thus, the evaluated limit is

\[ \boxed{1}. \]