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To solve the limit problem using the definition of the derivative, we can recognize that this expression represents the derivative of a function \( f(x) = \tan(x) \) at the point \( x \).

### The Definition of the Derivative
The derivative of a function \( f(x) \) with respect to \( x \) is defined as:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

### Applying to \( f(x) = \tan(x) \)

Given the limit:

\[
\lim_{h \to 0} \frac{\tan(x+h) - \tan(x)}{h}
\]

This is the definition of the derivative of \( f(x) = \tan(x) \). Therefore:

\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]

So, the limit evaluates to:

\[
\lim_{h \to 0} \frac{\tan(x+h) - \tan(x)}{h} = \sec^2(x)
\]

### Final Answer
The limit is \( \sec^2(x) \).

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How is the derivative of \( \tan(x) \) derived from first principles?
2. What is the derivative of \( \sec(x) \) using the definition of the derivative?
3. How can the chain rule be applied to find the derivative of more complex trigonometric functions?
4. What is the geometric interpretation of \( \sec^2(x) \)?
5. How would you use the definition of the derivative to find the derivative of \( \cot(x) \)?

**Tip:** The derivative of a trigonometric function often reveals useful identities that are key in solving more complex calculus problems.

Learn how to solve the limit problem using the definition of the derivative for tan(x). Understand the process of applying the limit definition to find the derivative of tan(x), resulting in sec^2(x). Suitable for college-level calculus students.

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