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We are given the following limit:

\[
\lim_{x \to 3} \frac{x^2 - 9x + 18}{x^2 - 10x + 21}
\]

### Step 1: Factor both the numerator and denominator

We can try factoring the quadratic expressions in both the numerator and the denominator.

#### Numerator:
\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\]

#### Denominator:
\[
x^2 - 10x + 21 = (x - 3)(x - 7)
\]

### Step 2: Simplify the expression
Now that we have factored both the numerator and denominator, we can rewrite the limit as:

\[
\lim_{x \to 3} \frac{(x - 3)(x - 6)}{(x - 3)(x - 7)}
\]

We can cancel the common factor \((x - 3)\) from both the numerator and denominator, assuming \(x \neq 3\). This simplifies the expression to:

\[
\lim_{x \to 3} \frac{x - 6}{x - 7}
\]

### Step 3: Substitute \(x = 3\)
Now, substitute \(x = 3\) into the simplified expression:

\[
\frac{3 - 6}{3 - 7} = \frac{-3}{-4} = \frac{3}{4}
\]

### Conclusion:
The limit exists, and its value is \( \frac{3}{4} \).

Would you like more details or further explanation on this? Here are some related questions:

1. How do you factor quadratic equations?
2. What is L'Hôpital's rule and when should it be applied in limits?
3. What does it mean when a limit does not exist?
4. What are common techniques for simplifying limits with indeterminate forms?
5. How does canceling terms work in simplifying limits?

**Tip:** Always attempt factoring first when evaluating limits of rational functions, as it often simplifies the problem.

Does lim (x -> 3) (x^2 - 9x + 18) / (x^2 - 10x + 21) exist?

Learn how to evaluate the limit of the rational function (x^2 - 9x + 18) / (x^2 - 10x + 21) as x approaches 3. This involves factoring quadratic expressions and simplifying the result. Suitable for high school students in Grades 10-12.

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