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We are tasked with calculating the following limit:

\[
\lim_{x \to \frac{\pi}{2}^{-}} \frac{\tan(x)}{\sec(x)}
\]

### Step-by-Step Solution:

1. **Rewrite using basic trigonometric identities**:
   Recall that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\). Substituting these into the expression:

   \[
   \frac{\tan(x)}{\sec(x)} = \frac{\frac{\sin(x)}{\cos(x)}}{\frac{1}{\cos(x)}} = \sin(x)
   \]

2. **Simplify the expression**:
   Now, the given limit reduces to:

   \[
   \lim_{x \to \frac{\pi}{2}^{-}} \sin(x)
   \]

3. **Evaluate the limit**:
   As \(x\) approaches \(\frac{\pi}{2}\) from the left, the value of \(\sin(x)\) approaches 1. Hence, the limit is:

   \[
   \lim_{x \to \frac{\pi}{2}^{-}} \sin(x) = 1
   \]

### Final Answer:
The value of the limit is \(1\).

---

Would you like more details or have any questions?

Here are five related questions:
1. What is the limit of \(\sec(x)\) as \(x\) approaches \(\frac{\pi}{2}\) from the left?
2. How does \(\tan(x)\) behave as \(x \to \frac{\pi}{2}^{-}\)?
3. Can the limit of \(\sin(x)\) ever exceed 1 as \(x \to \frac{\pi}{2}\)?
4. How does the limit change if the angle were \(\frac{\pi}{4}\) instead of \(\frac{\pi}{2}\)?
5. What is the derivative of \(\sin(x)\) and how could it be applied to limits involving trigonometric functions?

**Tip:** Always rewrite trigonometric expressions using identities to simplify limits before evaluating them.

Calculate the limit lim (x → π/2^−) (tg x / sec x).

Learn how to calculate the limit of tg x / sec x as x approaches π/2 from the left. This problem involves applying basic trigonometric identities and evaluating limits, ideal for students in Grades 11-12.

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