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?

---

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.

This step-by-step solution shows how to evaluate the limit lim/x→0 (sin x - tan x) / x^3 by applying Taylor series expansions for sin x and cos x. The solution involves simplifying the trigonometric expressions and using limit laws to find the final answer: -1/2. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation