Solve for the limit as x goes to 0 of sec(x)cot(x) analytically

We are asked to find the limit of the expression \( \sec(x) \cot(x) \) as \( x \) approaches 0:

\[
\lim_{x \to 0} \sec(x) \cot(x)
\]

### Step 1: Rewrite \( \sec(x) \) and \( \cot(x) \) in terms of basic trigonometric functions

We know that:

- \( \sec(x) = \frac{1}{\cos(x)} \)
- \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

Substituting these into the expression, we get:

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

### Step 2: Simplify the expression

Thus, the original expression reduces to:

\[
\frac{1}{\sin(x)}
\]

### Step 3: Take the limit as \( x \to 0 \)

We now need to find the limit:

\[
\lim_{x \to 0} \frac{1}{\sin(x)}
\]

Since \( \sin(x) \to 0 \) as \( x \to 0 \), the limit of \( \frac{1}{\sin(x)} \) approaches \( \infty \). Therefore:

\[
\lim_{x \to 0} \frac{1}{\sin(x)} = \infty
\]

### Conclusion:
The limit of \( \sec(x) \cot(x) \) as \( x \) approaches 0 is \( \infty \).

Would you like further details on this solution or have any questions?

Here are 5 related questions to consider:
1. What is the limit of \( \frac{1}{\tan(x)} \) as \( x \to 0 \)?
2. How does the behavior of trigonometric functions around 0 affect limits?
3. What is the significance of limits approaching infinity in real-world scenarios?
4. Can L'Hopital's Rule be applied to solve similar limits, and how?
5. What are some common limits involving \( \sec(x) \) and \( \cot(x) \)?

**Tip**: When dealing with trigonometric limits, rewriting the functions in simpler terms (like sine and cosine) often makes the limit easier to compute.

Explore the limit of sec(x)cot(x) as x approaches 0 with a detailed step-by-step solution. This problem covers trigonometric identities and limit theorems, making it suitable for undergraduate calculus students.

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