To prove that:

$\lim_{{x \to 3}} (x^2 - x) = 6,$

we'll follow these steps:

Step 1: Direct Substitution

The limit is asking what happens as $x$ approaches 3 for the expression $x^2 - x$. One method to evaluate limits is direct substitution if the function is continuous around the point $x = 3$.

Substitute $x = 3$ into the expression:

$x^2 - x = 3^2 - 3 = 9 - 3 = 6.$

Thus, the limit is:

$\lim_{{x \to 3}} (x^2 - x) = 6.$

Conclusion:

We have proven that:

$\lim_{{x \to 3}} (x^2 - x) = 6.$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you compute the limit if the function were not continuous?
2. What is the definition of a limit, and why is direct substitution valid here?
3. How can the limit be computed using epsilon-delta methods for more complex cases?
4. How does the limit change if the expression was $x^2 - 2x$?
5. What if $x$ approached a different value, such as 2? What would the new limit be?

Tip: Always check whether a function is continuous at the point of interest before using direct substitution to find the limit.