Step 1: Find the Maclaurin Series for $\sin^2(x)$

We will begin by using the known series expansion for $\cos(2x)$, which is:

$\cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \frac{(2x)^6}{6!} + \cdots$

This simplifies to:

$\cos(2x) = 1 - 2x^2 + \frac{2x^4}{3} - \frac{2x^6}{45} + \cdots$

We want to express $\sin^2(x)$. We can use the identity:

$\sin^2(x) = \frac{1 - \cos(2x)}{2}$

Substitute the Maclaurin series for $\cos(2x)$:

$\sin^2(x) = \frac{1 - \left( 1 - 2x^2 + \frac{2x^4}{3} - \cdots \right)}{2}$

Simplify this:

$\sin^2(x) = \frac{2x^2 - \frac{2x^4}{3} + \cdots}{2}$

$\sin^2(x) = x^2 - \frac{x^4}{3} + \cdots$

Step 2: Find the Limit $\lim_{x \to 0} \frac{\sin^2(x)}{x^2 - x^4}$

Now, we compute the limit:

$\lim_{x \to 0} \frac{\sin^2(x)}{x^2 - x^4}$

Using the Maclaurin series expansion for $\sin^2(x)$:

$\sin^2(x) = x^2 - \frac{x^4}{3} + \cdots$

So, the limit becomes:

$\lim_{x \to 0} \frac{x^2 - \frac{x^4}{3} + \cdots}{x^2 - x^4}$

Factor out $x^2$ from both the numerator and denominator:

$= \lim_{x \to 0} \frac{x^2 \left(1 - \frac{x^2}{3} + \cdots \right)}{x^2 (1 - x^2)}$

Canceling the common $x^2$ term:

$= \lim_{x \to 0} \frac{1 - \frac{x^2}{3} + \cdots}{1 - x^2}$

As $x \to 0$, both the numerator and denominator approach 1:

$\lim_{x \to 0} \frac{1 - \frac{x^2}{3}}{1 - x^2} = \frac{1}{1} = 1$

Final Answer:

$\lim_{x \to 0} \frac{\sin^2(x)}{x^2 - x^4} = 1$

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Do you need more details on any part of this process or have any questions?

Here are some related questions for further exploration:

1. How do you find the Maclaurin series for other trigonometric functions, like $\cos(x)$?
2. Can you derive the Maclaurin series for $\sin(x)$ and compare it with the series for $\sin^2(x)$?
3. What happens if you apply a higher-order approximation for $\sin^2(x)$ in the limit expression?
4. How do you expand $\sin(x)$ and $\cos(x)$ into series and use them for approximations in limits?
5. How would you derive a series for $\sin^3(x)$ using a similar method?

Tip: Always check the first few terms of your series expansion before using them for approximations in limits!