The limit expression shown in the image is:

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

To solve this limit, let's first simplify the expression.

Step 1: Combine the two fractions

To combine the fractions, we need a common denominator. The denominators are $x^2 - x$ and $x - 1$. We can factor $x^2 - x$ as:

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

So the expression becomes:

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

Step 2: Rewrite with a common denominator

Now, express both terms with the denominator $x(x - 1)$:

$\frac{1}{x(x - 1)} - \frac{1}{x - 1} = \frac{1}{x(x - 1)} - \frac{x}{x(x - 1)}$

This simplifies to:

$\frac{1 - x}{x(x - 1)}$

Step 3: Simplify the numerator

Notice that the numerator $1 - x$ can be rewritten as $-(x - 1)$, so the expression becomes:

$\frac{-(x - 1)}{x(x - 1)}$

Step 4: Cancel out common terms

We can cancel the $(x - 1)$ terms from the numerator and denominator, leaving us with:

$\frac{-1}{x}$

Step 5: Take the limit as $x \to 1$

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

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

Final Answer:

The limit is $-1$.

Would you like further explanation or assistance with any part of the solution?

Here are some related questions to explore:

1. How do you handle limits involving indeterminate forms like $\frac{0}{0}$?
2. What is the difference between a one-sided limit and a two-sided limit?
3. How can we apply L'Hôpital's Rule to simplify limits?
4. What are other common limit properties that help in simplifying expressions?
5. How would the result change if the limit were taken as $x \to 0$?

Tip: Always check for common factors that can simplify limits, especially when encountering indeterminate forms!