The given expression is:

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

Let's solve it step by step.

Step 1: Simplify the Denominator

Notice that the denominator $x^2 - 1$ can be factored:

$x^2 - 1 = (x - 1)(x + 1)$

Thus, the expression becomes:

$\lim_{{x \to 1}} \left( \frac{x(x + 1)}{(x - 1)(x + 1)} - \frac{2}{(x - 1)(x + 1)} \right)$

Step 2: Combine the Terms

Since both terms have the same denominator, we can combine them into a single fraction:

$\lim_{{x \to 1}} \frac{x(x + 1) - 2}{(x - 1)(x + 1)}$

Step 3: Simplify the Numerator

Now, simplify the numerator $x(x + 1) - 2$:

$x(x + 1) - 2 = x^2 + x - 2$

Therefore, the expression becomes:

$\lim_{{x \to 1}} \frac{x^2 + x - 2}{(x - 1)(x + 1)}$

Step 4: Factor the Numerator

Notice that $x^2 + x - 2$ can be factored as:

$x^2 + x - 2 = (x - 1)(x + 2)$

So the expression becomes:

$\lim_{{x \to 1}} \frac{(x - 1)(x + 2)}{(x - 1)(x + 1)}$

Step 5: Cancel the Common Factor

We can cancel the common factor of $(x - 1)$ from the numerator and the denominator:

$\lim_{{x \to 1}} \frac{x + 2}{x + 1}$

Step 6: Evaluate the Limit

Now, substitute $x = 1$ into the simplified expression:

$\frac{1 + 2}{1 + 1} = \frac{3}{2}$

Final Answer:

The value of the limit is:

$\boxed{\frac{3}{2}}$

Would you like more details, or do you have any further questions?

Here are five related questions you can explore:

1. What happens if we change the limit point from $x \to 1$ to another value?
2. How would the solution change if the denominator was not factorable?
3. What is the significance of factoring when solving limits involving polynomials?
4. How do you handle limits that result in indeterminate forms like $\frac{0}{0}$?
5. Can L'Hopital's Rule be used to solve this limit? If so, how?

Tip: Always check if a limit expression simplifies before directly substituting the value of $x$.