calculate the lim as n goes to 9 of : (x^2-81)/(6ln(x+8)) using series

To calculate the limit of \(\lim_{x \to 9} \frac{x^2 - 81}{6 \ln(x+8)}\) using series expansions, we will use a Taylor series expansion for both the numerator and denominator around \(x = 9\).

### Step-by-step approach:

#### Step 1: Simplify the expression
Notice that the numerator can be factored as:
\[
x^2 - 81 = (x - 9)(x + 9)
\]
This is useful because it helps us work with a simpler term as \(x \to 9\).

#### Step 2: Series expansion for \(\ln(x+8)\) around \(x = 9\)
We expand \(\ln(x+8)\) around \(x = 9\). Let \(x = 9 + h\), where \(h \to 0\). Then, \(x + 8 = 17 + h\). The series expansion of \(\ln(17 + h)\) around \(h = 0\) is:
\[
\ln(17 + h) = \ln(17) + \frac{h}{17} + O(h^2)
\]
Thus, for small \(h\),
\[
\ln(x + 8) \approx \ln(17) + \frac{x - 9}{17}
\]

#### Step 3: Approximation of the numerator
The numerator is \((x^2 - 81)\), which we factored as \((x - 9)(x + 9)\). For \(x\) near 9, this becomes:
\[
x^2 - 81 = (x - 9)(x + 9) \approx (x - 9) \times 18 = 18(x - 9)
\]

#### Step 4: Construct the limit
Now, we approximate both the numerator and denominator:
\[
\frac{x^2 - 81}{6 \ln(x + 8)} \approx \frac{18(x - 9)}{6 \left( \ln(17) + \frac{x - 9}{17} \right)}
\]
Simplifying further:
\[
\frac{18(x - 9)}{6\ln(17) + \frac{6(x - 9)}{17}} = \frac{3(x - 9)}{\ln(17) + \frac{x - 9}{17}}
\]

#### Step 5: Take the limit as \(x \to 9\)
As \(x \to 9\), \(x - 9 \to 0\), so the expression simplifies to:
\[
\lim_{x \to 9} \frac{3(x - 9)}{\ln(17) + \frac{x - 9}{17}} = \frac{0}{\ln(17)} = 0
\]

### Final Answer:
\[
\lim_{x \to 9} \frac{x^2 - 81}{6 \ln(x + 8)} = 0
\]

Let me know if you'd like further details or clarifications!

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Here are 8 related questions:

1. What is a Taylor series, and how is it used to approximate functions?
2. How can you apply the L'Hôpital's rule to find limits of indeterminate forms?
3. What is the significance of series expansion in calculus?
4. Can you explain why the logarithmic function is approximated by a series around a specific point?
5. How does factorization help in simplifying limit problems?
6. What is the importance of approximating functions near points of interest in calculus?
7. How would the result change if we used a different method, like L'Hôpital's rule?
8. What is the general form of a logarithmic series expansion?

**Tip:** For limits involving indeterminate forms, series expansions or L'Hôpital's rule are two powerful tools!

Learn how to calculate the limit of (x^2 - 81) / (6 ln(x+8)) using series expansions and Taylor series around x = 9. Understand the step-by-step approach to simplify the expression and find the limit effectively.

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