Determine lim(square root (9n^2+8n+12)-3n+18

We are tasked with determining the following limit:

\[
\lim_{{n \to \infty}} \left( \sqrt{9n^2 + 8n + 12} - 3n + 18 \right)
\]

### Step 1: Simplify the square root expression
As \( n \to \infty \), the dominant term inside the square root is \( 9n^2 \), so let's factor out \( n^2 \) from the square root expression:

\[
\sqrt{9n^2 + 8n + 12} = n\sqrt{9 + \frac{8}{n} + \frac{12}{n^2}}
\]

### Step 2: Approximate the square root
As \( n \to \infty \), the terms \( \frac{8}{n} \) and \( \frac{12}{n^2} \) approach zero. So, we can approximate the square root using a binomial expansion for large \( n \):

\[
\sqrt{9 + \frac{8}{n} + \frac{12}{n^2}} \approx 3 + \frac{4}{3n}
\]

Thus, we approximate the square root expression as:

\[
\sqrt{9n^2 + 8n + 12} \approx n \left( 3 + \frac{4}{3n} \right) = 3n + \frac{4}{3}
\]

### Step 3: Substitute the approximation into the limit
Now, substitute this approximation into the original limit expression:

\[
\lim_{{n \to \infty}} \left( \left( 3n + \frac{4}{3} \right) - 3n + 18 \right)
\]

Simplifying this expression:

\[
\lim_{{n \to \infty}} \left( \frac{4}{3} + 18 \right) = \frac{4}{3} + 18 = \frac{4}{3} + \frac{54}{3} = \frac{58}{3}
\]

### Final Answer:
\[
\lim_{{n \to \infty}} \left( \sqrt{9n^2 + 8n + 12} - 3n + 18 \right) = \frac{58}{3}
\]

Would you like more details on this solution, or do you have any other questions?

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Here are 5 related questions to explore:
1. How does the binomial expansion help in approximating square roots for large values of \( n \)?
2. What happens if we expand the square root to a higher order term?
3. How would the limit change if we altered the coefficients in the quadratic inside the square root?
4. What role does the constant 18 play in the final result?
5. How would you approach a similar limit problem with a cubic term?

**Tip:** Always look for the dominant term as \( n \to \infty \) to simplify limits involving polynomials.

Determine lim(square root (9n^2+8n+12)-3n+18)

Discover how to find the limit of sqrt(9n^2+8n+12)-3n+18 as n approaches infinity. This problem involves limits, asymptotic approximations, and binomial expansion for large n. Suitable for advanced high school or early college students.

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