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 can follow these steps:

### Step 1: Identify the limit expression
We start with the limit expression:
\[ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \]

### Step 2: Factor the numerator
The expression \( x^2 - 1 \) can be factored using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} \]

### Step 3: Simplify the expression
We can simplify the expression by canceling the \( (x - 1) \) terms in the numerator and the denominator, provided \( x \neq 1 \):
\[ \lim_{x \to 1} (x + 1) \]

### Step 4: Evaluate the limit
Now, we can directly substitute \( x = 1 \) into the simplified expression:
\[ 1 + 1 = 2 \]

### Conclusion
Thus, we find that:
\[ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2 \]