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The given problem is to evaluate the following limit:

\[
\lim_{{x \to 3}} \frac{x^2 + 2x + 1}{x - 2}
\]

### Step 1: Direct Substitution
First, let's check the limit by directly substituting \(x = 3\) into the expression.

\[
\frac{3^2 + 2(3) + 1}{3 - 2} = \frac{9 + 6 + 1}{1} = \frac{16}{1} = 16
\]

### Final Answer:
\[
\lim_{{x \to 3}} \frac{x^2 + 2x + 1}{x - 2} = 16
\]

Would you like further explanation or details on this process?

Here are five related questions you might want to explore:
1. What happens if the denominator was zero at the limit point?
2. How would you handle limits where direct substitution leads to \(0/0\)?
3. What techniques can you use for limits involving complex expressions?
4. Can limits always be evaluated using substitution?
5. What are the conditions for a limit to exist?

**Tip:** Always check if direct substitution leads to a defined value before applying other methods like factoring or L'Hôpital's rule.

Evaluate the limit lim (x^2 + 2x + 1)/(x - 2) as x approaches 3.

Learn how to evaluate the limit lim (x^2 + 2x + 1)/(x - 2) as x approaches 3 using direct substitution. Understand key calculus concepts and the application of limits to rational functions. Suitable for students in Grades 11-12.

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