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

We are tasked with finding the limit of the function $ f(x) = \frac{4x^3 + 1}{2x^2 + \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 denominator. Let's simplify the expression by factoring out the dominant term from both the numerator and denominator.

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

### Step 2: Factor out the dominant term

Let's factor out $ x^3 $ from both the numerator and the denominator.

For the **numerator** $ 4x^3 + 1 $:
\[
4x^3 + 1 = x^3(4 + \frac{1}{x^3})
\]

For the **denominator**, first simplify the square root $ \sqrt{16x^6 + 1} $. Factor out $ x^6 $ from inside the square root:
\[
\sqrt{16x^6 + 1} = |x^3|\sqrt{16 + \frac{1}{x^6}} = -x^3\sqrt{16 + \frac{1}{x^6}} \quad \text{(since $ x $ is negative)}
\]
Thus, the denominator becomes:
\[
2x^2 + \sqrt{16x^6 + 1} = x^3\left(\frac{2}{x} + \sqrt{16 + \frac{1}{x^6}}\right)
\]

### Step 3: Express the function in terms of large $ x $

Now, rewrite the function in simplified form:
\[
f(x) = \frac{x^3(4 + \frac{1}{x^3})}{x^3\left(\frac{2}{x} + \sqrt{16 + \frac{1}{x^6}}\right)}
\]

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

### 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, so the function simplifies to:
\[
f(x) = \frac{4}{\sqrt{16}} = \frac{4}{4} = 1
\]

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

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you evaluate limits when both numerator and denominator tend to infinity?
2. What happens when the square root term dominates the denominator in limits?
3. How can you handle limits involving negative infinity for rational functions?
4. How would this limit change if the powers of $ x $ in the numerator and denominator were different?
5. How do you apply L'Hopital's Rule to evaluate complex limits?
6. How do limits change if there are trigonometric functions involved?
7. How do horizontal asymptotes relate to limits at infinity?
8. How do you find limits involving polynomial and radical functions?

**Tip:** When evaluating limits at infinity, always identify the dominant terms, as they largely dictate the behavior of the function for large values of $ x $.

Explore the limit of the function f(x) = (4x^3 + 1) / (2x^2 + sqrt(16x^6 + 1)) as x approaches negative infinity. Understand the polynomial division and the behavior of square roots in limits. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation