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The problem asks to find and prove the limit:

\[
\lim_{{x \to 5}} \sqrt[4]{\frac{3x + 1}{3x - 1}}
\]

### Step 1: Direct Substitution

Let's first try direct substitution by plugging \( x = 5 \) into the given expression:

\[
\frac{3(5) + 1}{3(5) - 1} = \frac{15 + 1}{15 - 1} = \frac{16}{14} = \frac{8}{7}
\]

Thus, the expression simplifies to:

\[
\sqrt[4]{\frac{8}{7}}
\]

### Step 2: Final Limit Value

So, the limit is:

\[
\lim_{{x \to 5}} \sqrt[4]{\frac{3x + 1}{3x - 1}} = \sqrt[4]{\frac{8}{7}}
\]

This is the simplified form of the result. You can leave the answer as it is, or you can further approximate it numerically if needed.

Would you like a detailed breakdown of how this proof works or the numerical approximation? Let me know if you'd like additional information or clarifications.

### Further Related Questions:
1. How do we handle indeterminate forms in limits?
2. What happens if the denominator approaches zero in limits?
3. Can L'Hôpital's rule be applied in this problem?
4. How can we approximate the fourth root of a number using a calculator?
5. What is the importance of limits in understanding continuity?

#### Tip:
When evaluating limits, always try direct substitution first. If you get an indeterminate form like \( \frac{0}{0} \), consider algebraic simplifications or L'Hôpital's rule.

Find the limit: lim (x -> 5) of the fourth root of (3x + 1)/(3x - 1) and prove it.

Solve and prove the limit lim (x -> 5) of the fourth root of (3x + 1)/(3x - 1). This problem involves evaluating limits and simplifying radical expressions, ideal for high school students studying algebra and calculus.

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