Evaluate the following limits :
`Lim_( x to 1^(-)) (x^(2) -1)/(|x-1|)`

Answer

To evaluate the limit

\[ \lim_{x \to 1^-} \frac{x^2 - 1}{|x - 1|}, \]

we will follow these steps:

### Step 1: Identify the expression and the limit direction
We need to evaluate the limit as \( x \) approaches \( 1 \) from the left (denoted as \( 1^- \)). The expression is

\[ \frac{x^2 - 1}{|x - 1|}. \]

### Step 2: Simplify the absolute value
Since we are approaching \( 1 \) from the left, \( x < 1 \). Therefore, we can simplify the absolute value:

\[ |x - 1| = -(x - 1) = 1 - x. \]

### Step 3: Rewrite the limit
Substituting \( |x - 1| \) in the limit gives us:

\[ \lim_{x \to 1^-} \frac{x^2 - 1}{1 - x}. \]

### Step 4: Factor the numerator
The numerator \( x^2 - 1 \) can be factored using the difference of squares:

\[ x^2 - 1 = (x - 1)(x + 1). \]

### Step 5: Substitute the factorization into the limit
Now we can substitute this factorization into our limit:

\[ \lim_{x \to 1^-} \frac{(x - 1)(x + 1)}{1 - x}. \]

### Step 6: Simplify the expression
Notice that \( 1 - x = -(x - 1) \). Thus, we can rewrite the limit as:

\[ \lim_{x \to 1^-} \frac{(x - 1)(x + 1)}{-(x - 1)}. \]

### Step 7: Cancel the common terms
We can cancel \( (x - 1) \) from the numerator and the denominator (as long as \( x \neq 1 \)):

\[ \lim_{x \to 1^-} -(x + 1). \]

### Step 8: Evaluate the limit
Now we can directly substitute \( x = 1 \):

\[ -(1 + 1) = -2. \]

### Final Answer
Thus, the limit is

\[ \boxed{-2}. \]