To solve the limit

$\lim_{{x \to 0}} \frac{\sin x - \tan x}{x^3}$

we'll analyze the expression in the numerator, $\sin x - \tan x$, and rewrite it to make it easier to evaluate.

Step 1: Rewrite $\tan x$ in terms of $\sin x$ and $\cos x$

We know that $\tan x = \frac{\sin x}{\cos x}$, so we can rewrite the expression as: $\frac{\sin x - \frac{\sin x}{\cos x}}{x^3} = \frac{\sin x \left(1 - \frac{1}{\cos x}\right)}{x^3}$

Step 2: Simplify $1 - \frac{1}{\cos x}$

Using the identity $1 - \frac{1}{\cos x} = \frac{\cos x - 1}{\cos x}$, we can rewrite the expression as: $\frac{\sin x \cdot \frac{\cos x - 1}{\cos x}}{x^3} = \frac{\sin x \cdot (\cos x - 1)}{x^3 \cdot \cos x}$

Step 3: Expand $\sin x$ and $\cos x - 1$ using Taylor series

For small $x$, we can use the Taylor series expansions: $\sin x \approx x - \frac{x^3}{6} + O(x^5)$ $\cos x - 1 \approx -\frac{x^2}{2} + O(x^4)$

Substitute these approximations into the expression: $\frac{\left(x - \frac{x^3}{6}\right) \cdot \left(-\frac{x^2}{2}\right)}{x^3 \cdot \left(1 + O(x^2)\right)}$

Step 4: Simplify the expression

Multiplying out terms, we get: $\frac{\left(-\frac{x^3}{2} + \frac{x^5}{12}\right)}{x^3} = \frac{-\frac{x^3}{2} + \frac{x^5}{12}}{x^3} = -\frac{1}{2} + \frac{x^2}{12}$

Step 5: Take the limit as $x \to 0$

As $x \to 0$, the $\frac{x^2}{12}$ term goes to zero, leaving us with: $\lim_{{x \to 0}} \frac{\sin x - \tan x}{x^3} = -\frac{1}{2}$

Final Answer

$\boxed{-\frac{1}{2}}$

Would you like additional details on Taylor series expansion, or help with any questions about this process?

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1. What is the Taylor series and how does it help in evaluating limits?
2. How can you approximate $\sin x$ and $\cos x$ for small $x$?
3. Why do higher-order terms vanish when taking limits?
4. What other methods can be used for limits involving trigonometric functions?
5. How does Taylor expansion compare with L'Hôpital's Rule for solving limits?

Tip: For limits involving indeterminate forms like $\frac{0}{0}$, Taylor expansions can simplify the problem by approximating functions around the point of interest.