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.