Evaluate: `lim_(xrarr1) ((2)/(x^(2)-1)+(1)/(1-x))`

Options

A. `1/2`
B. `1/4`
C. `-1/2`
D. `-1/4`

Explanation

To evaluate the limit

\[ \lim_{x \to 1} \left( \frac{2}{x^2 - 1} + \frac{1}{1 - x} \right), \]

we can start by simplifying the expression.

### Step 1: Factor the denominator

The term \(x^2 - 1\) can be factored as:

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

So we rewrite the limit as:

\[ \lim_{x \to 1} \left( \frac{2}{(x - 1)(x + 1)} + \frac{1}{1 - x} \right). \]

### Step 2: Rewrite the second term

Notice that \(1 - x = -(x - 1)\). Therefore, we can rewrite the second term:

\[ \frac{1}{1 - x} = -\frac{1}{x - 1}. \]

Now the limit becomes:

\[ \lim_{x \to 1} \left( \frac{2}{(x - 1)(x + 1)} - \frac{1}{x - 1} \right). \]

### Step 3: Combine the fractions

To combine the two fractions, we need a common denominator, which is \((x - 1)(x + 1)\):

\[ \frac{2}{(x - 1)(x + 1)} - \frac{1(x + 1)}{(x - 1)(x + 1)} = \frac{2 - (x + 1)}{(x - 1)(x + 1)}. \]

This simplifies to:

\[ \frac{2 - x - 1}{(x - 1)(x + 1)} = \frac{1 - x}{(x - 1)(x + 1)}. \]

### Step 4: Simplify the expression

Notice that \(1 - x = -(x - 1)\), so we can rewrite the expression as:

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

### Step 5: Evaluate the limit

Now we can evaluate the limit as \(x\) approaches 1:

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

Thus, the final result is:

\[ \boxed{-\frac{1}{2}}. \]