\lim _{x\to 0}\frac{sin\left(2x\right)}{x}

To solve the limit \(\lim_{x \to 0} \frac{\sin(2x)}{x}\), let's proceed step-by-step using limit properties and trigonometric identities.

### Step-by-Step Solution

1. **Use Substitution:**
   Let's make a substitution to help simplify the limit. Set \(u = 2x\). As \(x \to 0\), \(u \to 0\) as well.

   Rewrite the expression in terms of \(u\):
   \[
   \lim_{x \to 0} \frac{\sin(2x)}{x} = \lim_{u \to 0} \frac{\sin(u)}{\frac{u}{2}}
   \]

2. **Simplify the Expression:**
   The expression \(\frac{\sin(u)}{\frac{u}{2}}\) can be simplified by multiplying the numerator and the denominator by 2:
   \[
   \lim_{u \to 0} \frac{\sin(u)}{\frac{u}{2}} = \lim_{u \to 0} \frac{2\sin(u)}{u}
   \]

3. **Apply the Sine Limit Identity:**
   We know from the standard limit identity that \(\lim_{u \to 0} \frac{\sin(u)}{u} = 1\). 

   Applying this identity:
   \[
   \lim_{u \to 0} \frac{2\sin(u)}{u} = 2 \lim_{u \to 0} \frac{\sin(u)}{u} = 2 \times 1 = 2
   \]

### Conclusion

Thus, the limit is:
\[
\lim_{x \to 0} \frac{\sin(2x)}{x} = 2
\]

Would you like more details on this solution or have any questions?

Here are some related questions to further your understanding:

1. What is the limit of \(\lim_{x \to 0} \frac{\sin(3x)}{x}\)?
2. How does the substitution method help in evaluating limits involving trigonometric functions?
3. Can we apply L'Hopital's Rule to solve \(\lim_{x \to 0} \frac{\sin(2x)}{x}\)? What would be the result?
4. What are some other common limits involving trigonometric functions?
5. How would the solution change if the function was \(\lim_{x \to 0} \frac{\tan(2x)}{x}\)?

**Tip:** Always consider using trigonometric identities and substitution when simplifying limits involving trigonometric functions.

Learn how to solve the limit lim(x → 0) sin(2x)/x with a detailed step-by-step solution using limit properties and trigonometric identities. Understand the application of substitution and the sine limit identity theorem. Suitable for advanced high school students and anyone studying 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