https://file.aiquickdraw.com/mathbot/file/1726913174977ru5k9scf.jpg

The expression in the image appears to be the limit of the following function as \( x \) approaches 5:

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

Let's evaluate this limit step by step:

1. **Substitute \( x = 5 \) into the function**:
   \[
   \frac{3(5) + 1}{3(5) - 1} = \frac{15 + 1}{15 - 1} = \frac{16}{14} = \frac{8}{7}
   \]

2. **Now take the fourth root**:
   \[
   \sqrt[4]{\frac{8}{7}}
   \]

This is the simplified limit expression. The exact result is \( \sqrt[4]{\frac{8}{7}} \).

Would you like to explore this in more detail, or have any other questions?

Here are five related questions to deepen your understanding:

1. What is the difference between evaluating a limit and finding a derivative?
2. How do you handle limits that result in indeterminate forms (e.g., \( \frac{0}{0} \))?
3. What is the significance of higher-order roots, like the fourth root, in calculus?
4. How can L'Hopital's Rule be applied to limit problems?
5. What is the general form of a continuous function, and why is continuity important in limits?

**Tip:** When evaluating limits, substituting the value directly into the function is the first step. If it leads to indeterminate forms, other techniques like factoring or L'Hopital's Rule might be necessary!

Evaluate the limit as x approaches 5 for the function (3x + 1) / (3x - 1) under the fourth root.

Learn how to evaluate the limit of (3x + 1) / (3x - 1) as x approaches 5, under a fourth root. This step-by-step guide explains the substitution method and the limit theorem to help students in Grades 11-12 understand advanced limit problems.

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