$$ \lim_{x\rarr-\infty}\frac{\left(\sqrt{3x^2+4x^4}\right)}{\left(\sqrt{x^4-x^2}\right)} $$

To solve the limit:

\[
\lim_{x \to -\infty} \frac{\sqrt{3x^2 + 4x^4}}{\sqrt{x^4 - x^2}},
\]
we will follow these steps:

### Step 1: Simplify the expressions inside the square roots.
- Focus on the highest power of $x$ in both the numerator and the denominator.
- For the numerator, the expression is $ \sqrt{3x^2 + 4x^4} $. Since $4x^4$ is the dominant term, factor out $x^4$:
  \[
  \sqrt{3x^2 + 4x^4} = \sqrt{x^4(4 + \frac{3}{x^2})} = |x^2|\sqrt{4 + \frac{3}{x^2}} = x^2\sqrt{4 + \frac{3}{x^2}}.
  \]

- For the denominator, the expression is $ \sqrt{x^4 - x^2} $. Again, factor out $x^4$ from inside the square root:
  \[
  \sqrt{x^4 - x^2} = \sqrt{x^4(1 - \frac{1}{x^2})} = |x^2|\sqrt{1 - \frac{1}{x^2}} = x^2\sqrt{1 - \frac{1}{x^2}}.
  \]

### Step 2: Rewrite the limit expression.
Now the limit becomes:
\[
\lim_{x \to -\infty} \frac{x^2\sqrt{4 + \frac{3}{x^2}}}{x^2\sqrt{1 - \frac{1}{x^2}}}.
\]

### Step 3: Simplify the expression.
Since $x^2$ appears in both the numerator and the denominator, they cancel out:
\[
\lim_{x \to -\infty} \frac{\sqrt{4 + \frac{3}{x^2}}}{\sqrt{1 - \frac{1}{x^2}}}.
\]

### Step 4: Evaluate the limit as $x \to -\infty$.
As $x \to -\infty$, $\frac{3}{x^2} \to 0$ and $\frac{1}{x^2} \to 0$. So the expression simplifies to:
\[
\frac{\sqrt{4 + 0}}{\sqrt{1 - 0}} = \frac{\sqrt{4}}{\sqrt{1}} = \frac{2}{1} = 2.
\]

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

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How do limits involving infinity affect the highest powers in polynomials?
2. What role do dominant terms play in simplifying expressions with limits?
3. How do you handle limits when both numerator and denominator involve square roots?
4. Can you explain why factoring out the dominant term helps in solving limits?
5. What are the key differences when evaluating limits at positive versus negative infinity?

**Tip:** When dealing with limits involving square roots and infinity, always focus on the dominant term (the highest power of $x$) for simplification.

Learn how to compute the limit as x approaches negative infinity for the expression (sqrt(3x^2+4x^4))/(sqrt(x^4-x^2)). This detailed solution involves algebraic simplification, square roots, and understanding the behavior of polynomials at infinity. Suitable for advanced high school and college students.

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