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use l hopitals to do this but dont use any extra identities

The provided expression is:

\[
\lim_{{x \to 0}} \frac{x^2}{{\sin^2 x}}
\]

To solve this limit using L'Hopital's Rule, let's follow these steps:

1. **Apply L'Hopital's Rule**: This rule is used when a limit results in an indeterminate form like \(\frac{0}{0}\). In this case, both \(x^2\) and \(\sin^2(x)\) approach 0 as \(x \to 0\), giving an indeterminate form.

2. Differentiate the numerator and denominator:
   - Derivative of \(x^2\) with respect to \(x\) is \(2x\).
   - Derivative of \(\sin^2(x)\) with respect to \(x\) is \(2\sin(x)\cos(x)\) (using the chain rule).

3. The new limit becomes:
   \[
   \lim_{{x \to 0}} \frac{2x}{2\sin(x)\cos(x)}
   \]

4. Simplify the expression:
   \[
   \lim_{{x \to 0}} \frac{x}{\sin(x)\cos(x)}
   \]

   Since \(\frac{x}{\sin(x)} \to 1\) as \(x \to 0\) and \(\cos(0) = 1\), the limit becomes:

   \[
   1 \times 1 = 1
   \]

Thus, the limit is:

\[
\boxed{1}
\]

Would you like further details, or do you have any questions? 

Here are five related questions:
1. What are the conditions for applying L'Hopital's Rule?
2. Can L'Hopital's Rule be applied more than once if the indeterminate form persists?
3. What happens if we differentiate both the numerator and denominator again?
4. How do we differentiate trigonometric functions like \(\sin^2(x)\)?
5. What are other common indeterminate forms where L'Hopital's Rule is useful?

**Tip**: Always check the indeterminate form before applying L'Hopital’s Rule.

Use L'Hopital's Rule to evaluate the limit of (x^2)/(sin^2(x)) as x approaches 0.

Learn how to solve the limit of (x^2)/(sin^2(x)) as x approaches 0 using L'Hopital's Rule. This detailed step-by-step solution includes the application of derivatives and key theorems such as the limit of sin(x)/x. Ideal for calculus students.

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