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 limit
We need to evaluate the limit as \( x \) approaches 5 from the left side (denoted as \( 5^- \)). This means that \( x \) is slightly less than 5.

### Step 2: Substitute \( x \)
Since we are approaching 5 from the left, we can express \( x \) as \( x = 5 - h \), where \( h \) is a small positive number (i.e., \( h \to 0^+ \)).

### Step 3: Rewrite the expression
Substituting \( x = 5 - h \) into the limit gives us:
\[ \frac{(5 - h) - 5}{| (5 - h) - 5 |} = \frac{-h}{| -h |} \]

### Step 4: Simplify the absolute value
The absolute value \( | -h | \) is equal to \( h \) because \( h \) is positive. Therefore, we can rewrite the expression as:
\[ \frac{-h}{h} \]

### Step 5: Simplify the fraction
Now, we simplify the fraction:
\[ \frac{-h}{h} = -1 \]

### Step 6: Conclude the limit
Thus, we find that:
\[ \lim_{x \to 5^-} \frac{x - 5}{|x - 5|} = -1 \]

### Final Answer
The limit is \( -1 \).

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