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Insert the numerical value of the second expression here

We are asked to find the limit \( L_2 \), which is expressed as:

\[
L_2 = \lim_{t \to 0} \cos\left( \frac{\pi}{\sqrt{7 - 3 \sec(2t)}} \right)
\]

### Step-by-step solution:

1. **Behavior of \( \sec(2t) \) as \( t \to 0 \):**
   \[
   \sec(2t) = \frac{1}{\cos(2t)}
   \]
   As \( t \to 0 \), \( \cos(2t) \to 1 \), and thus \( \sec(2t) \to 1 \).

2. **Substitute into the expression:**
   As \( t \to 0 \), \( \sec(2t) \to 1 \), so the term \( 7 - 3\sec(2t) \) becomes:
   \[
   7 - 3(1) = 4
   \]

3. **Simplifying the square root:**
   Now, we have:
   \[
   \sqrt{7 - 3 \sec(2t)} \to \sqrt{4} = 2
   \]

4. **Substitute this into the cosine expression:**
   The expression inside the cosine becomes:
   \[
   \frac{\pi}{2}
   \]

5. **Evaluate the cosine:**
   \[
   \cos\left( \frac{\pi}{2} \right) = 0
   \]

### Conclusion:
\[
L_2 = 0
\]

Let me know if you would like further details or clarifications!

### Related Questions:
1. What is the behavior of \( \sec(2t) \) as \( t \to 0 \)?
2. How do trigonometric functions simplify when \( t \to 0 \)?
3. Why does \( \cos(\frac{\pi}{2}) = 0 \)?
4. Can you find similar limits involving trigonometric functions?
5. How do small-angle approximations help in solving limits?

### Tip:
Always check the behavior of trigonometric functions like \( \sec(2t) \) at boundary values (e.g., \( t \to 0 \)) for simplified results in limits!

Find the limit L_2 = lim_{t -> 0} cos(π / √(7 - 3 sec(2t))).

This calculus problem explores finding the limit of the function cos(π / √(7 - 3 sec(2t))) as t approaches 0. The step-by-step solution covers small-angle approximations and the behavior of trigonometric functions in limit processes. Suitable for college-level students.

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