Evaluate : `lim_(xto5^(-))(x+5)/(|x+5|)`

Answer

To evaluate the limit \( \lim_{x \to 5^-} \frac{x + 5}{|x + 5|} \), we can follow these steps:

### Step 1: Understand the behavior of \( |x + 5| \) as \( x \) approaches 5 from the left.
Since we are approaching 5 from the left (denoted as \( 5^- \)), we need to determine the sign of \( x + 5 \) in this region.

As \( x \) approaches 5 from the left, \( x \) will be slightly less than 5, which means:
\[ x + 5 < 10. \]
Thus, \( x + 5 \) is positive when \( x \) is close to 5 from the left.

### Step 2: Write the expression for \( |x + 5| \).
Since \( x + 5 \) is positive in this case, we have:
\[ |x + 5| = x + 5. \]

### Step 3: Substitute into the limit expression.
Now, we can substitute this into our limit:
\[ \lim_{x \to 5^-} \frac{x + 5}{|x + 5|} = \lim_{x \to 5^-} \frac{x + 5}{x + 5}. \]

### Step 4: Simplify the expression.
The expression simplifies to:
\[ \lim_{x \to 5^-} 1 = 1. \]

### Conclusion:
Thus, the limit evaluates to:
\[ \lim_{x \to 5^-} \frac{x + 5}{|x + 5|} = 1. \]