limit x tends to zero ax-sin3x/x^3 = b then 2b/a is

We are tasked with finding \( \frac{2b}{a} \) given the limit:

\[
\lim_{x \to 0} \frac{ax - \sin(3x)}{x^3} = b.
\]

### Step 1: Expand \( \sin(3x) \) using the Taylor series
The Taylor series expansion of \( \sin(3x) \) around \( x = 0 \) is:

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

Thus,

\[
\sin(3x) = 3x - \frac{27x^3}{6} + \mathcal{O}(x^5).
\]

### Step 2: Substitute the expansion into the limit
Now, substitute the Taylor series expansion of \( \sin(3x) \) into the limit expression:

\[
\frac{ax - \sin(3x)}{x^3} = \frac{ax - \left( 3x - \frac{27x^3}{6} + \mathcal{O}(x^5) \right)}{x^3}.
\]

Simplify the numerator:

\[
ax - \sin(3x) = ax - \left(3x - \frac{27x^3}{6} \right) = (a - 3)x + \frac{27x^3}{6}.
\]

Thus, the limit becomes:

\[
\frac{(a - 3)x + \frac{27x^3}{6}}{x^3}.
\]

### Step 3: Simplify the limit expression
Now, split the fraction:

\[
\frac{(a - 3)x}{x^3} + \frac{\frac{27x^3}{6}}{x^3} = \frac{a - 3}{x^2} + \frac{27}{6}.
\]

As \( x \to 0 \), the term \( \frac{a - 3}{x^2} \) blows up unless \( a = 3 \). Therefore, for the limit to exist, we must have \( a = 3 \).

### Step 4: Find \( b \)
With \( a = 3 \), the limit becomes:

\[
\lim_{x \to 0} \frac{\frac{27}{6}}{1} = \frac{27}{6} = 4.5.
\]

Thus, \( b = 4.5 \).

### Step 5: Find \( \frac{2b}{a} \)
Now, we compute:

\[
\frac{2b}{a} = \frac{2 \times 4.5}{3} = \frac{9}{3} = 3.
\]

Thus, \( \frac{2b}{a} = 3 \).

### Final Answer:
\[
\boxed{3}.
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the Taylor series expansion of \( \sin(x) \)?
2. How do you compute limits involving trigonometric functions?
3. What happens when you apply L'Hopital's Rule to indeterminate forms?
4. How can you find the value of a constant that makes a limit exist?
5. What are common techniques for handling limits with small angles?

**Tip**: The Taylor series is a powerful tool to approximate functions near specific points, particularly for limit problems.

Learn how to solve the limit problem lim(x→0) (ax - sin(3x)) / x^3 = b using Taylor series expansion. This solution explains the detailed steps, including expanding sin(3x), simplifying the limit expression, and calculating 2b/a. Suitable for undergraduate students.

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