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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.

Prove that lim(x^2 - x) as x approaches 3 is 6.

Learn how to prove that the limit of x^2 - x as x approaches 3 equals 6 using direct substitution. This problem covers key concepts in limits, polynomial functions, and continuity, suitable for students in Grades 10-12.

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