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The image you provided is a mathematical problem involving limits. The problem asks to find the limit:

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

The solution proceeds step by step. Here's a breakdown:

### Step 1: Expressing the equation
We are given the limit expression:
\[
\frac{x^2 + 3x}{\sqrt{2x^4 + 3 - 4x}}
\]
The goal is to simplify this expression for \( x \to \infty \).

### Step 2: Factoring out dominant terms
- In the numerator \( x^2 + 3x \), the term \( x^2 \) dominates as \( x \to \infty \).
- In the denominator, the dominant term inside the square root is \( 2x^4 \) because as \( x \) grows large, other terms become negligible.

We factor out \( x^2 \) from both the numerator and denominator to make the expression easier to evaluate as \( x \to \infty \):

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

### Step 3: Simplification
As \( x \to \infty \), terms involving \( \frac{1}{x}, \frac{1}{x^3}, \) and \( \frac{1}{x^4} \) tend to zero. Therefore, the expression simplifies to:

\[
\frac{1}{\sqrt{2}} 
\]

### Step 4: Conclusion
Thus, the limit is:

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

Would you like more details on any specific step or the rules applied here?

### Further Questions:
1. Why do we focus on the dominant terms as \( x \to \infty \)?
2. What happens to smaller order terms as \( x \) approaches infinity?
3. How does the square root affect the terms in the denominator?
4. Could this limit be evaluated using L'Hopital's rule instead?
5. What does the result \( \frac{1}{\sqrt{2}} \) imply about the growth rates of the numerator and denominator?

### Tip:
When evaluating limits as \( x \to \infty \), always identify and simplify dominant terms, as smaller ones become negligible!

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

Learn how to find the limit of (x^2 + 3x) / sqrt(2x^4 + 3 - 4x) as x approaches infinity. This problem involves key calculus concepts such as limits, asymptotic behavior, and theorems related to the limit of functions. Suitable for high school or introductory college calculus students.

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