https://file.aiquickdraw.com/mathbot/file/17285993217837yesmz4b.jpg

The expression given in the image is:

\[
\lim_{x \to 2} \frac{x^2 + 5x + 6}{x^2 + 4x}
\]

Let's break it down:

1. **Substitute \(x = 2\) into the expression:**

   \[
   \frac{2^2 + 5(2) + 6}{2^2 + 4(2)} = \frac{4 + 10 + 6}{4 + 8} = \frac{20}{12} = \frac{5}{3}
   \]

So, the limit is:

\[
\boxed{\frac{5}{3}}
\]

Would you like further details or have any questions about this solution?

Here are 5 related questions for further practice:

1. What is the limit of \(\lim_{x \to 3} \frac{x^2 + 2x + 1}{x^2 + 3x + 2} \)?
2. How do you handle limits that result in an indeterminate form like \( \frac{0}{0} \)?
3. Can you find the limit of a function as \( x \to \infty \) for rational expressions?
4. What is L'Hopital's Rule and when is it applied in limit problems?
5. How do you factor quadratic expressions to simplify limit problems?

**Tip**: Always check if direct substitution works first when solving limits. If it results in an indeterminate form, try factoring, canceling, or using other techniques like L'Hopital's Rule.

Evaluate the limit as x approaches 2 for the expression (x^2 + 5x + 6) / (x^2 + 4x).

Learn how to evaluate the limit of the rational function (x^2 + 5x + 6) / (x^2 + 4x) as x approaches 2. This problem demonstrates the use of substitution and limit laws, suitable for students in Grades 11-12.

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