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The problem is asking to evaluate the limit:

\[
\lim_{x \to 0} \frac{\tan(2x)}{\sec(x)}
\]

### Step 1: Express the secant function
Recall that \(\sec(x)\) is the reciprocal of the cosine function:
\[
\sec(x) = \frac{1}{\cos(x)}
\]

Thus, the given expression becomes:
\[
\lim_{x \to 0} \frac{\tan(2x)}{\frac{1}{\cos(x)}} = \lim_{x \to 0} \tan(2x) \cdot \cos(x)
\]

### Step 2: Analyze the limit
We need to find:
\[
\lim_{x \to 0} \tan(2x) \cdot \cos(x)
\]

As \(x \to 0\):
- \(\tan(2x) \to 2x\) because \(\tan(\theta) \approx \theta\) for small \(\theta\), and here \(\theta = 2x\), so \(\tan(2x) \approx 2x\).
- \(\cos(x) \to 1\) since \(\cos(0) = 1\).

Thus, the limit simplifies to:
\[
\lim_{x \to 0} 2x \cdot 1 = 0
\]

### Conclusion:
The limit of the given expression as \(x\) approaches 0 is **0**.

Would you like further details or have any other questions?

### Related Questions:
1. What is the derivative of \(\tan(2x)\)?
2. How do you find limits involving trigonometric functions?
3. What is L'Hopital's Rule and when is it applied in limit problems?
4. How do small angle approximations work for trigonometric functions?
5. How does the behavior of \(\cos(x)\) and \(\tan(x)\) differ as \(x\) approaches zero?

**Tip:** Small angle approximations are often useful when evaluating limits of trigonometric functions around zero!

Which of the following is true for the limit lim x→0 tan(2x)/sec(x)?

Explore how to solve the limit lim x→0 tan(2x)/sec(x) using trigonometric identities and small angle approximations. This step-by-step solution is suitable for students in Grades 11-12 or early college studying calculus.

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