The problem asks to find the following limit:

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

Step 1: Try direct substitution

Let's first substitute $x = 1$ directly into the function:

$\frac{(1)^2 + (1) - 2}{(1)^2 - (1) + 1} = \frac{1 + 1 - 2}{1 - 1 + 1} = \frac{0}{1} = 0$

The result is a well-defined number (0), not an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Conclusion:

The limit exists, and its value is:

$\boxed{0}$

Would you like more details, or do you have any questions? Here are 5 related questions you could explore:

1. What is the limit of $\lim_{x \to 2} \frac{x^2 + x - 2}{x^2 - x + 1}$?
2. How would you solve a limit when you encounter indeterminate forms like $\frac{0}{0}$?
3. Can you use L'Hopital's Rule for solving limits of this kind? If so, how?
4. What is the geometric interpretation of limits involving rational functions?
5. How does factoring help in solving limits of rational functions?

Tip:

Always check if direct substitution can solve the limit before applying more advanced techniques like factoring or L'Hopital's Rule.