We are tasked with finding the limit of the expression:

$\lim_{{x \to -3}} \frac{3x^2 + 3x - 36}{x^2}$

Step 1: Plug in $x = -3$

Let's first substitute $x = -3$ directly into the expression and check if we can evaluate the limit:

$\frac{3(-3)^2 + 3(-3) - 36}{(-3)^2} = \frac{3(9) + 3(-3) - 36}{9} = \frac{27 - 9 - 36}{9} = \frac{-18}{9} = -2$

Thus, the limit is:

$\lim_{{x \to -3}} \frac{3x^2 + 3x - 36}{x^2} = -2$

Final Answer:

The limit as $x$ approaches $-3$ is $-2$.

Would you like to see more details or any clarifications?

Here are some related questions you might find helpful:

1. What are some common methods for solving limits that result in indeterminate forms?
2. How do you find limits when substitution results in undefined expressions?
3. Can you explain how L'Hopital's Rule applies in more complex limits?
4. What are the differences between limits at infinity and finite limits?
5. How does factoring help in solving limits of rational functions?

Tip: Always check if substituting the limit value directly leads to a valid result before applying other limit-solving techniques.