Step 1: Maclaurin Series for $\sin(\sin(x))$

The Maclaurin series expansion is essentially the Taylor series expansion centered at $x = 0$. Let us find the first three terms of $\sin(\sin(x))$.

1. Start with the series for $\sin(x)$: $\sin(x) = x - \frac{x^3}{6} + \frac{x^5}{120} + \dots$ Substitute $\sin(x)$ into itself: $\sin(\sin(x)) = \sin\left(x - \frac{x^3}{6} + \frac{x^5}{120} + \dots\right).$

2. Expand $\sin(y)$ for $y = x - \frac{x^3}{6} + \dots$: Using the series for $\sin(y)$, we have: $\sin(y) = y - \frac{y^3}{6} + \dots$ Substitute $y = x - \frac{x^3}{6}$: $\sin(y) = \left(x - \frac{x^3}{6}\right) - \frac{\left(x - \frac{x^3}{6}\right)^3}{6}.$

3. Simplify the terms: For the first three terms:

   • Linear term: $x$
   • Cubic term: $-\frac{x^3}{6}$
   • Higher-order terms ($\left(x - \frac{x^3}{6}\right)^3$): $\left(x - \frac{x^3}{6}\right)^3 = x^3 - \frac{x^9}{6^3} + \dots \approx x^3$ After expanding, this leads to additional contributions to higher-order terms.

Thus, up to $O(x^5)$: $\sin(\sin(x)) \approx x - \frac{x^3}{6}.$

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Step 2: Simplify the given expression

We now analyze: $\lim_{x \to 0} \frac{x - \sin(\sin(x))}{x^3}.$

Using the expansion of $\sin(\sin(x)) \approx x - \frac{x^3}{6}$: $x - \sin(\sin(x)) = x - \left(x - \frac{x^3}{6}\right) = \frac{x^3}{6}.$

Thus: $\frac{x - \sin(\sin(x))}{x^3} = \frac{\frac{x^3}{6}}{x^3} = \frac{1}{6}.$

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Final Answer:

$\lim_{x \to 0} \frac{x - \sin(\sin(x))}{x^3} = \frac{1}{6}.$

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Would you like to see more details on the Maclaurin series expansion or on any step above? Let me know!

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Related Questions:

1. What is the general formula for the Maclaurin series of $\sin(x)$?
2. How does substitution affect the terms of a Taylor series?
3. What is the significance of $\lim_{x \to 0}$ in this context?
4. How can we compute higher-order terms in $\sin(\sin(x))$?
5. What are other series expansions that are useful in calculus?

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Tip: When substituting into a Taylor series, carefully consider the order of terms required to avoid over-approximating or truncating too soon.