Math Problem Statement
Solution
The problem requires finding the limit:
Step-by-Step Solution
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Analyze the Denominator: For small values of , we can use the Taylor series expansion for around : Therefore,
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Rewrite the Limit: Substitute the approximation for :
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Simplify the Expression: As , the term becomes negligible compared to , so we approximate:
Final Answer
Would you like further clarification on any step?
Related Questions:
- How can we use Taylor series to approximate other trigonometric functions around zero?
- Why does the higher-order term become negligible in this problem?
- How can L'Hôpital's Rule be applied to evaluate limits involving indeterminate forms?
- What is the general process for finding limits of ratios that go to indeterminate forms?
- How would the solution change if the power in the numerator was different?
Tip:
Taylor series expansions are useful for approximating complex functions near specific points, especially for calculating limits and simplifying expressions.
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Math Problem Analysis
Mathematical Concepts
Limits
Taylor Series Expansion
Trigonometric Functions
Formulas
\( \tan(x) \approx x + \frac{x^3}{3} + O(x^5) \)
\( \lim_{{x \to 0}} \frac{x^3}{-\frac{x^3}{3}} = -3 \)
Theorems
Taylor Series Expansion
Suitable Grade Level
Grades 11-12
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