what is the limit of (5x^3 + 8)/(sqrt(36n^6 + 4)) as x approaches infinity

To find the limit of 

\[
\lim_{x \to \infty} \frac{5x^3 + 8}{\sqrt{36x^6 + 4}},
\]

we’ll follow these steps:

### Step 1: Simplify the Expression
Since \( x \) approaches infinity, we want to analyze the terms with the highest power of \( x \) in both the numerator and the denominator, as they will dominate the behavior of the function.

1. **Numerator:** The highest power of \( x \) in the numerator is \( 5x^3 \).
2. **Denominator:** The highest power of \( x \) in the denominator is under a square root. Expanding the square root term, we get \( \sqrt{36x^6 + 4} \approx \sqrt{36x^6} = 6x^3 \) for large \( x \).

### Step 2: Rewrite the Expression in Terms of the Highest Power
Now, we can rewrite the limit as:

\[
\lim_{x \to \infty} \frac{5x^3 + 8}{\sqrt{36x^6 + 4}} = \lim_{x \to \infty} \frac{5x^3 + 8}{6x^3}.
\]

### Step 3: Divide Both Terms by \( x^3 \)
To simplify further, divide both the numerator and denominator by \( x^3 \):

\[
= \lim_{x \to \infty} \frac{5 + \frac{8}{x^3}}{6}.
\]

### Step 4: Take the Limit
As \( x \to \infty \), the term \( \frac{8}{x^3} \) approaches 0. So, we are left with:

\[
= \frac{5 + 0}{6} = \frac{5}{6}.
\]

### Final Answer
\[
\lim_{x \to \infty} \frac{5x^3 + 8}{\sqrt{36x^6 + 4}} = \frac{5}{6}.
\]

Would you like further details on any step?

Here are some related questions:
1. How would the answer change if the numerator were \( 7x^3 + 10 \)?
2. What is the behavior of this function as \( x \to -\infty \)?
3. How can this problem be solved using L'Hôpital's Rule?
4. What if the denominator were \( \sqrt{49x^6 + 9} \) instead?
5. How does the limit change if we add a \( x^2 \) term in the numerator?

**Tip:** In limits involving polynomials and square roots, identify the highest-power terms first to simplify calculations.

What is the limit of (5x^3 + 8)/(sqrt(36x^6 + 4)) as x approaches infinity?

Explore how to find the limit of (5x^3 + 8)/(sqrt(36x^6 + 4)) as x approaches infinity. This step-by-step solution covers limits at infinity, polynomial functions, and asymptotic analysis to simplify complex expressions. Ideal for high school students in Grades 11-12.

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