The limit problem you've given is:

\[
\lim_{x \to -3^+} \frac{x^2 + 4}{x^2 - 9}
\]

Let's solve this step by step.

### Step 1: Factor the denominator
The denominator \(x^2 - 9\) can be factored using the difference of squares:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

Now the expression becomes:

\[
\lim_{x \to -3^+} \frac{x^2 + 4}{(x - 3)(x + 3)}
\]

### Step 2: Check the limit as \(x \to -3^+\)
When \(x \to -3^+\), the behavior of each part of the fraction must be considered:

- The numerator \(x^2 + 4\) approaches \(9 + 4 = 13\).
- The factor \(x - 3\) approaches \(-6\) because \(x\) is slightly greater than \(-3\) from the right.
- The factor \(x + 3\) approaches \(0\), but since \(x \to -3^+\), \(x + 3\) is slightly positive.

### Step 3: Analyze the behavior of the limit
Since the numerator approaches 13 (a finite number) and the denominator approaches 0, we analyze the sign of the denominator. The term \(x + 3\) is positive when approaching from the right of \(-3\), and \(x - 3\) is negative. This means the denominator is approaching 0 from the negative side.

Thus, the limit can be interpreted as:

\[
\lim_{x \to -3^+} \frac{13}{\text{small negative number}} = -\infty
\]

### Final Answer:
\[
\lim_{x \to -3^+} \frac{x^2 + 4}{x^2 - 9} = -\infty
\]

Would you like more details on any step, or do you have any questions?

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Here are 5 related questions for further practice:
1. \(\lim_{x \to 3^-} \frac{x^2 + 4}{x^2 - 9}\)
2. \(\lim_{x \to -3} \frac{x^2 - 4}{x^2 - 9}\)
3. \(\lim_{x \to 0} \frac{x^2 + 1}{x^2 - 1}\)
4. \(\lim_{x \to -3^+} \frac{x^2 + 9}{x^2 - 9}\)
5. \(\lim_{x \to 2^+} \frac{x^2 - 4}{x^2 - 1}\)

**Tip:** Always check the sign of the denominator when it approaches 0 to determine whether the limit tends toward \(+\infty\) or \(-\infty\).

Explore the step-by-step solution for evaluating the limit lim x → -3+ of (x^2 + 4)/(x^2 - 9). This problem involves factoring a difference of squares and analyzing the behavior of limits. Ideal for high school students in Grades 11-12 studying advanced algebra and calculus concepts.

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