Find the limit as x approaches 0, (x-3 sin(x))^2 / 10x^2

To find the limit

\[
\lim_{x \to 0} \frac{(x - 3\sin(x))^2}{10x^2}
\]

we'll follow these steps:

### Step 1: Use the series expansion of \(\sin(x)\)
The Taylor series expansion of \(\sin(x)\) around \(x = 0\) is:

\[
\sin(x) = x - \frac{x^3}{6} + O(x^5)
\]

Thus, for small \(x\):

\[
3\sin(x) = 3\left(x - \frac{x^3}{6} + O(x^5)\right) = 3x - \frac{x^3}{2} + O(x^5)
\]

### Step 2: Subtract \(x\) from \(3\sin(x)\)
Now subtract \(3\sin(x)\) from \(x\):

\[
x - 3\sin(x) = x - \left(3x - \frac{x^3}{2} + O(x^5)\right) = -2x + \frac{x^3}{2} + O(x^5)
\]

### Step 3: Square the result
We now square the expression \(x - 3\sin(x)\):

\[
(x - 3\sin(x))^2 = \left(-2x + \frac{x^3}{2} + O(x^5)\right)^2 = (-2x)^2 + 2(-2x)\left(\frac{x^3}{2}\right) + \left(\frac{x^3}{2}\right)^2 + O(x^7)
\]

This simplifies to:

\[
(x - 3\sin(x))^2 = 4x^2 + O(x^4)
\]

### Step 4: Substitute into the limit
Now, substitute the expression for \((x - 3\sin(x))^2\) into the original limit:

\[
\lim_{x \to 0} \frac{(x - 3\sin(x))^2}{10x^2} = \lim_{x \to 0} \frac{4x^2 + O(x^4)}{10x^2}
\]

Simplifying the expression:

\[
\lim_{x \to 0} \frac{4x^2}{10x^2} = \frac{4}{10} = \frac{2}{5}
\]

### Final Answer:
\[
\boxed{\frac{2}{5}}
\]

Would you like more details on any of the steps, or do you have further questions?

Here are five related questions for further exploration:
1. What is the Taylor series expansion of \(\cos(x)\) around \(x = 0\)?
2. How do higher-order terms affect the limit calculation in this type of problem?
3. Can the limit of this function be solved using L'Hôpital's Rule? Why or why not?
4. How would the limit change if the numerator involved \((x - 2\sin(x))^2\)?
5. What happens to the limit if you multiply the entire function by \(x\)?

**Tip:** When dealing with small \(x\), approximations via Taylor series often simplify complex trigonometric expressions.

Find the limit as x approaches 0, (x - 3 sin(x))^2 / 10x^2

This page provides a detailed solution to the limit problem: Find the limit as x approaches 0 of (x - 3 sin(x))^2 / 10x^2. The solution includes the use of Taylor series expansion and explores key concepts in calculus and asymptotic behavior. Ideal for undergraduate calculus students.

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