Math Problem Statement
Limit analisis
Solution
The given image shows the mathematical expression:
To solve this limit, let’s analyze it step by step.
Step 1: Use approximations for small values of
As , we can use the standard approximations for sine and tangent:
This simplifies the expression to:
Step 2: Conclusion
Thus, the limit evaluates to:
Would you like further clarification, or details on any specific step?
Follow-up Questions:
- What is L'Hopital's Rule and how could it apply to this limit?
- How are small-angle approximations useful for limits involving trigonometric functions?
- Can you explain why as ?
- How does the tangent approximation work for very small angles?
- How would this problem change if the limit were ?
Tip: When dealing with limits involving trigonometric functions, small-angle approximations like and 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)
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