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The given problem is to evaluate the limit:

\[
\lim_{x \to \infty} \frac{\sqrt[8]{8x^9 + 3x^4 + 1} + \sqrt[5]{x^{10} + x^7 + 1} + 10}{\sqrt[4]{x^4 + x^2 + 1} + \sqrt[6]{x^{12} + x^2 + 1} - 10}.
\]

To solve this limit as \( x \to \infty \), we follow these steps:

### Step 1: Identify Dominant Terms

- For the term \(\sqrt[8]{8x^9 + 3x^4 + 1}\), the dominant term is \( 8x^9 \).
- For the term \(\sqrt[5]{x^{10} + x^7 + 1}\), the dominant term is \( x^{10} \).
- For the term \(\sqrt[4]{x^4 + x^2 + 1}\), the dominant term is \( x^4 \).
- For the term \(\sqrt[6]{x^{12} + x^2 + 1}\), the dominant term is \( x^{12} \).

### Step 2: Simplify Each Term by Extracting Dominant Powers

1. \(\sqrt[8]{8x^9 + 3x^4 + 1} \approx (8x^9)^{1/8} = 8^{1/8}x^{9/8}\).
2. \(\sqrt[5]{x^{10} + x^7 + 1} \approx (x^{10})^{1/5} = x^2\).
3. \(\sqrt[4]{x^4 + x^2 + 1} \approx (x^4)^{1/4} = x\).
4. \(\sqrt[6]{x^{12} + x^2 + 1} \approx (x^{12})^{1/6} = x^2\).

### Step 3: Simplify the Limit Expression

The simplified form of the expression becomes:

\[
\frac{8^{1/8}x^{9/8} + x^2 + 10}{x + x^2 - 10}.
\]

### Step 4: Divide by the Highest Power of \(x\) in the Denominator

The highest power in the denominator is \( x^2 \). So, divide every term by \( x^2 \):

\[
\frac{\frac{8^{1/8}x^{9/8}}{x^2} + \frac{x^2}{x^2} + \frac{10}{x^2}}{\frac{x}{x^2} + \frac{x^2}{x^2} - \frac{10}{x^2}}.
\]

Simplify:

\[
\frac{\frac{8^{1/8}}{x^{7/8}} + 1 + \frac{10}{x^2}}{\frac{1}{x} + 1 - \frac{10}{x^2}}.
\]

### Step 5: Evaluate the Limit as \( x \to \infty \)

As \( x \to \infty \):

- \(\frac{8^{1/8}}{x^{7/8}} \to 0\).
- \(\frac{10}{x^2} \to 0\).
- \(\frac{1}{x} \to 0\).
- \(\frac{10}{x^2} \to 0\).

Thus, the expression simplifies to:

\[
\frac{0 + 1 + 0}{0 + 1 - 0} = \frac{1}{1} = 1.
\]

### Final Answer:

\[
\lim_{x \to \infty} \frac{\sqrt[8]{8x^9 + 3x^4 + 1} + \sqrt[5]{x^{10} + x^7 + 1} + 10}{\sqrt[4]{x^4 + x^2 + 1} + \sqrt[6]{x^{12} + x^2 + 1} - 10} = 1.
\]

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Would you like further details or have any questions about this solution?

### Related Questions:
1. How do you find the dominant term in a radical expression?
2. What is the importance of dividing by the highest power of \(x\) in limits?
3. How do you handle infinity in limits involving radicals?
4. What is the role of simplification in evaluating complex limits?
5. How do fractional exponents affect the behavior of functions as \(x \to \infty\)?

### Tip:
When evaluating limits involving radicals, always identify and simplify the dominant terms first to make the calculations manageable!

Evaluate the limit: lim_{x \to \infty} \frac{\sqrt[8]{8x^9 + 3x^4 + 1} + \sqrt[5]{x^{10} + x^7 + 1} + 10}{\sqrt[4]{x^4 + x^2 + 1} + \sqrt[6]{x^{12} + x^2 + 1} - 10}.

Learn how to evaluate a limit involving complex radicals as x approaches infinity. This step-by-step guide simplifies the dominant terms in the expression to help solve the problem. Suitable for students in undergraduate calculus.

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