For the rational function
g(x) = x^2 − 36/x^3 − 216
,
evaluate each of the following limits: limit as x approaches 6

To evaluate the limit of the rational function \( g(x) = \frac{x^2 - 36}{x^3 - 216} \) as \( x \) approaches 6, we will follow these steps:

1. **Substitute \( x = 6 \) into the function:**
   \[
   g(6) = \frac{6^2 - 36}{6^3 - 216} = \frac{36 - 36}{216 - 216} = \frac{0}{0}
   \]
   This results in an indeterminate form, \( \frac{0}{0} \), which suggests that we need to simplify the function.

2. **Factor both the numerator and the denominator:**
   - The numerator \( x^2 - 36 \) is a difference of squares:
     \[
     x^2 - 36 = (x - 6)(x + 6)
     \]
   - The denominator \( x^3 - 216 \) is a difference of cubes:
     \[
     x^3 - 216 = (x - 6)(x^2 + 6x + 36)
     \]

3. **Rewrite the function using the factored forms:**
   \[
   g(x) = \frac{(x - 6)(x + 6)}{(x - 6)(x^2 + 6x + 36)}
   \]
   We can cancel the common factor \( x - 6 \) from the numerator and the denominator:
   \[
   g(x) = \frac{x + 6}{x^2 + 6x + 36}
   \]

4. **Evaluate the limit by substituting \( x = 6 \) into the simplified function:**
   \[
   \lim_{x \to 6} g(x) = \lim_{x \to 6} \frac{x + 6}{x^2 + 6x + 36}
   \]
   Substitute \( x = 6 \):
   \[
   \lim_{x \to 6} \frac{6 + 6}{6^2 + 6 \cdot 6 + 36} = \frac{12}{36 + 36 + 36} = \frac{12}{108} = \frac{1}{9}
   \]

Thus, the limit of \( g(x) \) as \( x \) approaches 6 is \( \boxed{\frac{1}{9}} \).

Would you like more details or have any questions?

Here are some related questions you might want to ask:

1. How do you factor polynomials in general?
2. Can you explain the difference of squares and difference of cubes formulas?
3. What are other methods to evaluate limits besides factoring?
4. How do you handle limits that result in indeterminate forms like \(\frac{0}{0}\)?
5. Can you explain L'Hôpital's Rule and when to use it?

**Tip:** When evaluating limits resulting in an indeterminate form, always consider factoring, simplifying, or using special limit laws like L'Hôpital's Rule.

Learn how to evaluate the limit of the rational function g(x) = (x^2 - 36) / (x^3 - 216) as x approaches 6. This problem involves understanding limits, rational functions, and factoring techniques. Suitable for high school students in Grades 11-12.

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