Evaluate the limit: limit, start subscript, x, right arrow, 0, end subscript, start fraction, 1, plus, 3, sine, left parenthesis, x, right parenthesis, minus, cosine, left parenthesis, x, right parenthesis, divided by, 9, x, end fraction*x*→0lim​9*x*1+3sin(*x*)−cos(*x*)​

To evaluate the limit:

\[
\lim_{{x \to 0}} \frac{1 + 3\sin(x) - \cos(x)}{9x}
\]

we'll proceed by expanding \(\sin(x)\) and \(\cos(x)\) around \(x = 0\) using their Taylor series approximations:

1. **Taylor Series Expansions** (around \(x = 0\)):
   - \(\sin(x) \approx x - \frac{x^3}{6} + O(x^5)\)
   - \(\cos(x) \approx 1 - \frac{x^2}{2} + O(x^4)\)

2. **Substitute into the Expression**:
   Substitute these expansions into \(1 + 3\sin(x) - \cos(x)\):

   \[
   1 + 3\left(x - \frac{x^3}{6}\right) - \left(1 - \frac{x^2}{2}\right)
   \]

3. **Simplify**:
   Expanding and combining terms, we get:

   \[
   1 + 3x - \frac{x^3}{2} - 1 + \frac{x^2}{2} = 3x + \frac{x^2}{2} - \frac{x^3}{2}
   \]

4. **Form the Fraction**:
   Substitute back into the limit:

   \[
   \lim_{{x \to 0}} \frac{3x + \frac{x^2}{2} - \frac{x^3}{2}}{9x}
   \]

   Simplify by dividing each term in the numerator by \(9x\):

   \[
   = \lim_{{x \to 0}} \left(\frac{3x}{9x} + \frac{x^2}{18x} - \frac{x^3}{18x}\right) = \lim_{{x \to 0}} \left(\frac{1}{3} + \frac{x}{18} - \frac{x^2}{18}\right)
   \]

5. **Take the Limit**:
   As \(x \to 0\), the terms \(\frac{x}{18}\) and \(\frac{x^2}{18}\) go to 0, leaving:

   \[
   \frac{1}{3}
   \]

So, the limit is:

\[
\lim_{{x \to 0}} \frac{1 + 3\sin(x) - \cos(x)}{9x} = \frac{1}{3}
\]

---

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### Related Questions
1. How would we evaluate \(\lim_{{x \to 0}} \frac{\sin(x) - x}{x^3}\)?
2. What is the result of \(\lim_{{x \to 0}} \frac{\cos(x) - 1}{x^2}\)?
3. How can we apply L'Hôpital's Rule to similar limits involving trigonometric functions?
4. What happens to \(\lim_{{x \to 0}} \frac{1 - \cos(x)}{x}\)?
5. How do Taylor series approximations help in simplifying limits as \(x \to 0\)?

### Tip
Using Taylor series expansions is very effective for evaluating limits of functions as \(x \to 0\) when direct substitution or standard rules like L'Hôpital's Rule might not immediately simplify the expression.

Evaluate the limit: limit, start subscript, x, right arrow, 0, end subscript, start fraction, 1, plus, 3, sine, left parenthesis, x, right parenthesis, minus, cosine, left parenthesis, x, right parenthesis, divided by, 9, x, end fraction

Explore the detailed solution to evaluating the limit lim x→0 (1 + 3sin(x) - cos(x)) / 9x using Taylor series expansions for sin(x) and cos(x). This solution demonstrates trigonometric limits and series expansion techniques, ideal for undergraduate calculus students.

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