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объясни, как это решить

Let's analyze and solve the given limit problem step by step. The expression is:

\[
\lim_{x \to +\infty} \frac{(7x + \sqrt{5x})\sqrt{4x^2 + 6x + 2}}{\sqrt[3]{3x + 6} - \sqrt[4]{64x^8 + 5x^4 - 2x}}.
\]

### Step 1: Analyze the highest powers of \(x\)
When dealing with limits as \(x \to \infty\), the highest powers of \(x\) will dominate the behavior of each part of the expression. So let's identify the highest powers of \(x\) in both the numerator and the denominator.

#### Numerator:
- The term \(7x + \sqrt{5x}\) has a leading term of \(7x\) because \( \sqrt{5x} = \sqrt{5} \cdot x^{1/2} \), which grows slower than \(7x\).
- The term \(\sqrt{4x^2 + 6x + 2}\) has a leading term of \(\sqrt{4x^2} = 2x\), because \(4x^2\) is the highest degree term.

Thus, the leading behavior of the numerator is approximately:
\[
(7x)(2x) = 14x^2.
\]

#### Denominator:
- The term \(\sqrt[3]{3x + 6}\) has a leading term of \(\sqrt[3]{3x} = (3x)^{1/3} = x^{1/3} \cdot 3^{1/3}\).
- The term \(\sqrt[4]{64x^8 + 5x^4 - 2x}\) has a leading term of \(\sqrt[4]{64x^8} = (64x^8)^{1/4} = 2x^2\), because \(64x^8\) dominates.

Thus, the leading behavior of the denominator is approximately:
\[
x^{1/3} - 2x^2.
\]

### Step 2: Simplify the leading terms
Now that we have the leading terms, we can approximate the limit expression with these dominant terms:
\[
\frac{14x^2}{x^{1/3} - 2x^2}.
\]

### Step 3: Factor and divide by the highest power of \(x\)
The highest power of \(x\) in the denominator is \(x^2\), so we divide both the numerator and the denominator by \(x^2\):

\[
\frac{14x^2}{x^{1/3} - 2x^2} = \frac{14}{x^{1/3}/x^2 - 2} = \frac{14}{x^{-5/3} - 2}.
\]

As \(x \to \infty\), the term \(x^{-5/3}\) tends to 0. Therefore, the limit becomes:

\[
\lim_{x \to +\infty} \frac{14}{-2} = -7.
\]

### Final Answer:
The value of the limit is \(-7\).

Would you like more details or have any questions?

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Here are 5 related questions for practice:
1. What happens if the highest degree in the numerator was larger than the denominator?
2. How would you compute the limit if \(x\) approaches a negative infinity?
3. What role do radical and root functions play in limit behavior for large \(x\)?
4. How does dividing by the highest power of \(x\) simplify the process?
5. Can you solve a limit where both numerator and denominator approach infinity?

**Tip:** When calculating limits as \(x\) approaches infinity, always focus on the highest degree terms—lower degree terms become insignificant in comparison.

This problem demonstrates how to solve a limit involving a rational function with radicals and higher powers of x as x approaches infinity. The solution explores key concepts such as dominant terms, simplifying expressions, and dividing by the highest power of x. Suitable for university-level calculus students.

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