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The given limit expression is:

\[
\lim_{x \to +\infty} \frac{x^2 - 8}{\sqrt{3x^4 + x}}
\]

To solve this limit, let's analyze it step by step:

### Step 1: Simplify by identifying dominant terms
As \( x \to +\infty \), the dominant term in the numerator is \( x^2 \), and in the denominator, it is \( \sqrt{3x^4} = x^2\sqrt{3} \).

Thus, we can approximate:

\[
\lim_{x \to +\infty} \frac{x^2 - 8}{\sqrt{3x^4 + x}} \approx \lim_{x \to +\infty} \frac{x^2}{\sqrt{3x^4}} = \lim_{x \to +\infty} \frac{x^2}{x^2\sqrt{3}} = \frac{1}{\sqrt{3}}
\]

### Step 2: More rigorous analysis
To be more precise, let's factor out the highest power of \(x\) from both the numerator and the denominator. We do this by factoring \(x^2\) from the numerator and \(x^2\) (inside the square root) from the denominator:

\[
= \lim_{x \to +\infty} \frac{x^2(1 - \frac{8}{x^2})}{x^2\sqrt{3 + \frac{1}{x^3}}}
\]

Now, cancel out the \(x^2\) terms:

\[
= \lim_{x \to +\infty} \frac{1 - \frac{8}{x^2}}{\sqrt{3 + \frac{1}{x^3}}}
\]

### Step 3: Apply the limit
As \( x \to +\infty \), both \( \frac{8}{x^2} \to 0 \) and \( \frac{1}{x^3} \to 0 \). Therefore, the limit simplifies to:

\[
= \frac{1 - 0}{\sqrt{3 + 0}} = \frac{1}{\sqrt{3}}
\]

### Final Answer:
\[
\boxed{\frac{1}{\sqrt{3}}}
\]

Would you like more details or explanations on any of the steps? 

Here are some related questions to explore this topic further:

1. How would the result change if the numerator were \(x^3\) instead of \(x^2\)?
2. What happens if the limit tends to negative infinity instead of positive infinity?
3. How would we approach this problem if we had \(x^5\) in the denominator instead of \(x^4\)?
4. Can this limit be solved using L'Hôpital's Rule? If so, how?
5. How does the inclusion of small terms (like \( \frac{8}{x^2} \)) affect the behavior of the limit?

**Tip:** When solving limits as \(x \to \infty\), focusing on dominant terms helps simplify the process significantly.

Evaluate the limit: lim (x -> +∞) (x^2 - 8) / (sqrt(3x^4 + x))

Learn how to evaluate the limit of (x^2 - 8) / sqrt(3x^4 + x) as x approaches infinity. This problem involves using dominant term analysis and simplifying expressions step by step. Suitable for students in Grades 11-12.

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