Math Problem Statement

Limit analisis

Limit analysis for the expression \(\lim_{{x \to 0}} \frac{x^2}{\sin(x) \cdot \tan(2x)}\).

Solution

The given image shows the mathematical expression:

limx0x2sin(x)tan(2x)\lim_{{x \to 0}} \frac{x^2}{\sin(x) \cdot \tan(2x)}

To solve this limit, let’s analyze it step by step.

Step 1: Use approximations for small values of xx

As x0x \to 0, we can use the standard approximations for sine and tangent:

  • sin(x)x\sin(x) \approx x
  • tan(2x)2x\tan(2x) \approx 2x

This simplifies the expression to:

x2x2x=x22x2=12\frac{x^2}{x \cdot 2x} = \frac{x^2}{2x^2} = \frac{1}{2}

Step 2: Conclusion

Thus, the limit evaluates to:

limx0x2sin(x)tan(2x)=12\lim_{{x \to 0}} \frac{x^2}{\sin(x) \cdot \tan(2x)} = \frac{1}{2}

Would you like further clarification, or details on any specific step?

Follow-up Questions:

  1. What is L'Hopital's Rule and how could it apply to this limit?
  2. How are small-angle approximations useful for limits involving trigonometric functions?
  3. Can you explain why sin(x)x\sin(x) \approx x as x0x \to 0?
  4. How does the tangent approximation work for very small angles?
  5. How would this problem change if the limit were xx \to \infty?

Tip: When dealing with limits involving trigonometric functions, small-angle approximations like sin(x)x\sin(x) \approx x and tan(x)x\tan(x) \approx x are highly useful.

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Math Problem Analysis

Mathematical Concepts

Limit
Trigonometric approximations
Small-angle approximations

Formulas

\sin(x) \approx x
\tan(2x) \approx 2x

Theorems

Small-angle approximations
Limit theorems

Suitable Grade Level

Undergraduate calculus or advanced high school (Grades 11-12)