$$  $$find the limit approaching negative infinity of the function $$ f\left(x\right)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}} $$

We are tasked with finding the limit of the function:

\[
f(x) = \frac{4x^3 + 1}{2x^3 + \sqrt{16x^6 + 1}}
\]

as $ x $ approaches negative infinity.

### Step 1: Simplify the expression for large $ |x| $
When $ x $ approaches negative infinity, the higher powers of $ x $ will dominate both the numerator and the denominator. Let's factor out the highest power of $ x $ from both the numerator and denominator.

- **Numerator**: The dominant term is $ 4x^3 $.
- **Denominator**: The dominant term inside the square root is $ 16x^6 $, which simplifies to $ 4x^3 $.

### Step 2: Factor out $ x^3 $

#### Numerator:
\[
4x^3 + 1 = x^3\left(4 + \frac{1}{x^3}\right)
\]

#### Denominator:
The term $ \sqrt{16x^6 + 1} $ can be factored as:
\[
\sqrt{16x^6 + 1} = |x^3|\sqrt{16 + \frac{1}{x^6}} = -x^3 \sqrt{16 + \frac{1}{x^6}} \quad \text{(since $ x $ is negative)}
\]
Thus, the denominator becomes:
\[
2x^3 + \sqrt{16x^6 + 1} = x^3\left(2 + \sqrt{16 + \frac{1}{x^6}}\right)
\]

### Step 3: Rewrite the function
Now, rewrite the function $ f(x) $ as:
\[
f(x) = \frac{x^3\left(4 + \frac{1}{x^3}\right)}{x^3\left(2 + \sqrt{16 + \frac{1}{x^6}}\right)}
\]

Cancel out $ x^3 $ from both the numerator and denominator:
\[
f(x) = \frac{4 + \frac{1}{x^3}}{2 + \sqrt{16 + \frac{1}{x^6}}}
\]

### Step 4: Take the limit as $ x \to -\infty $
As $ x \to -\infty $, the terms $ \frac{1}{x^3} $ and $ \frac{1}{x^6} $ approach 0. The function simplifies to:
\[
f(x) = \frac{4}{2 + \sqrt{16}} = \frac{4}{2 + 4} = \frac{4}{6} = \frac{2}{3}
\]

### Final Answer:
\[
\lim_{x \to -\infty} \frac{4x^3 + 1}{2x^3 + \sqrt{16x^6 + 1}} = \frac{2}{3}
\]

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you approach limits at infinity for different types of functions?
2. How do horizontal asymptotes relate to limits at infinity?
3. What happens when the powers of $ x $ in the numerator and denominator are different?
4. How does the square root term affect the behavior of the function as $ x \to \infty $?
5. How would the limit change if the leading terms in the numerator and denominator were different?
6. How do you apply L'Hopital's Rule to limits involving indeterminate forms?
7. What is the limit if both numerator and denominator approach infinity at different rates?
8. How can you find limits involving exponential and logarithmic functions?

**Tip:** For limits at infinity, focus on the dominant terms with the highest powers of $ x $ in both the numerator and denominator to simplify the expression.

Explore how to find the limit of the function f(x) = (4x^3 + 1) / (2x^3 + sqrt(16x^6 + 1)) as x approaches negative infinity. Understand the mathematical steps involving dominant terms, factorization, and simplification techniques. Suitable for college-level students studying calculus and limits.

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