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The given limit is:

\[
\lim_{x \to \frac{\pi}{6}} \sqrt{\csc^2 x + 5 \sqrt{3} \tan x}
\]

### Step-by-Step Solution:

1. **Evaluate \(\csc^2 x\) and \(\tan x\) at \(x = \frac{\pi}{6}\):**
   - \(\sin \frac{\pi}{6} = \frac{1}{2}\), so:
     \[
     \csc \frac{\pi}{6} = \frac{1}{\sin \frac{\pi}{6}} = 2 \quad \text{and} \quad \csc^2 \frac{\pi}{6} = 4
     \]
   - \(\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}\)

2. **Substitute the values into the expression:**
   \[
   \lim_{x \to \frac{\pi}{6}} \sqrt{4 + 5 \sqrt{3} \cdot \frac{1}{\sqrt{3}}}
   \]
   Simplifying the term inside the square root:
   \[
   4 + 5 \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 4 + 5 = 9
   \]

3. **Simplify the square root:**
   \[
   \sqrt{9} = 3
   \]

Thus, the limit is:

\[
\boxed{3}
\]

---

Do you want more details or have any questions?

Here are 5 related questions to expand this topic:
1. What is the general form of the trigonometric identity involving \(\csc^2 x\)?
2. How can you apply limits to more complex trigonometric functions?
3. What are the key steps in evaluating limits involving radicals and trigonometric functions?
4. How would this problem change if the angle were different, such as \(x \to \frac{\pi}{4}\)?
5. How does the derivative of trigonometric functions help in solving limits?

**Tip:** When solving trigonometric limits, simplifying functions using known values for angles like \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\) often leads to quicker solutions.

lim (x -> π/6) sqrt(csc^2(x) + 5√3 * tan(x))

Learn how to evaluate the limit of sqrt(csc^2(x) + 5√3 * tan(x)) as x approaches π/6. This problem involves trigonometric identities, square roots, and the application of limits. Suitable for high school and early college students.

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