The image shows the step-by-step solution of a limit problem. Let's break it down for you.

The expression to solve is:

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

Step 1: Factor the numerator and denominator

The first transformation is factoring both the numerator and denominator. Starting with the given expression:

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

We can simplify the numerator as:

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

And the denominator can be factored as:

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

Thus, the expression becomes:

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

Step 2: Factor the numerator

Now, the numerator $x^2 + x$ can be factored as:

$x(x + 1)$

Thus, the expression becomes:

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

Step 3: Simplify the expression

At this point, you can cancel out the common factor $x$ from the numerator and denominator, assuming $x \neq 0$:

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

Step 4: Evaluate the limit

Now we can directly substitute $x = 1$ into the simplified expression:

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

However, because the denominator $x - 1$ tends to 0 as $x$ approaches 1, the limit does not exist in the traditional sense. Instead, we can consider the limit as $x \to 1^+$ and $x \to 1^-$.

1. When $x \to 1^+$ (from the right): The denominator $x - 1$ approaches 0 from the positive side, so the limit is $+\infty$.

2. When $x \to 1^-$ (from the left): The denominator $x - 1$ approaches 0 from the negative side, so the limit is $-\infty$.

Hence, there is no finite limit as $x \to 1$, and the limit diverges to infinity depending on the direction of approach.

Conclusion

The solution seems to involve simplifying the given expression step by step and examining the limit carefully. The final result should take into account the divergence of the limit as $x \to 1$. The image, however, concludes with $3$, which could be a typo or part of a different final form of the problem not shown fully.

Would you like to explore further details, or do you have specific questions?

Here are 5 related questions you might find interesting:

1. What happens when a limit approaches infinity in a more complex function?
2. How can we evaluate limits that involve trigonometric functions?
3. What is the L'Hôpital's Rule, and when is it applicable in evaluating limits?
4. How do limits apply in defining continuity and differentiability in calculus?
5. How can you use algebraic manipulation techniques to simplify complex rational functions?

Tip: When solving limit problems, always check if factoring or algebraic manipulation can simplify the expression before applying direct substitution.