To prove that \( \frac{\tan(x)}{x} = 1 \), we need to examine the statement more closely.

### Step 1: Understanding the claim

The equation \( \frac{\tan(x)}{x} = 1 \) suggests that \( \tan(x) = x \). This is **not true in general** for all values of \( x \). In fact, \( \tan(x) \) and \( x \) are different functions, and they do not satisfy this equation universally.

However, this equation holds **approximately** when \( x \) approaches 0. In other words, the limit of \( \frac{\tan(x)}{x} \) as \( x \to 0 \) is 1. Let's prove this.

### Step 2: Use of Limit

We aim to prove:

\[
\lim_{x \to 0} \frac{\tan(x)}{x} = 1
\]

To do this, we can apply L'Hôpital's Rule or use the Taylor expansion of \( \tan(x) \) around \( x = 0 \).

### Step 3: Taylor Expansion of \( \tan(x) \)

The Taylor series expansion of \( \tan(x) \) around \( x = 0 \) is:

\[
\tan(x) = x + \frac{x^3}{3} + O(x^5)
\]

For small values of \( x \), we can approximate \( \tan(x) \) as:

\[
\tan(x) \approx x
\]

### Step 4: Applying the Approximation

Using this approximation:

\[
\frac{\tan(x)}{x} \approx \frac{x}{x} = 1
\]

Thus, as \( x \to 0 \), \( \frac{\tan(x)}{x} \to 1 \).

### Step 5: Conclusion

We have shown that:

\[
\lim_{x \to 0} \frac{\tan(x)}{x} = 1
\]

This means the original statement \( \frac{\tan(x)}{x} = 1 \) is approximately true for very small \( x \). However, it is not true for all values of \( x \).

### Summary

The equation \( \frac{\tan(x)}{x} = 1 \) holds only **in the limit** as \( x \to 0 \). Otherwise, \( \tan(x) \) and \( x \) are different functions.

---

Do you have any questions or want more details on this? Here are five related questions you might find helpful:

1. What are other trigonometric limits similar to \( \lim_{x \to 0} \frac{\tan(x)}{x} = 1 \)?
2. How does L'Hôpital's Rule apply to solving limits like this?
3. Can we derive the Taylor series expansion of \( \tan(x) \) in more detail?
4. How can we use small-angle approximations in trigonometry?
5. What is the difference between \( \tan(x) \) and other trigonometric functions when approaching 0?

**Tip:** Remember that many trigonometric identities and limits hold true only under specific conditions, such as near certain values like \( x = 0 \).

Explore the proof that tan(x)/x approaches 1 as x tends to 0 using limits and Taylor series expansion. This step-by-step solution covers key trigonometric concepts, small-angle approximations, and the application of L'Hôpital's Rule. Ideal for students in Grades 11-12 studying pre-calculus or calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation