Math Problem Statement
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Solution
To solve this limit problem:
we'll use limit properties and trigonometric approximations for small . Here is the detailed solution:
Step 1: Expand Trigonometric Functions
For small values of , we use the following approximations:
Substitute these approximations into the limit:
Step 2: Simplify the Expression
Simplify the above expression:
The terms cancel out:
Final Answer
The result of the limit is , which corresponds to option (c).
Conclusion
Thus, the correct answer is:
c.
Would you like further details or have any other questions?
Here are 5 related questions that expand on the topic:
- How do you derive the small-angle approximations for trigonometric functions?
- What other methods can be used to solve limits involving trigonometric functions?
- How can L'Hôpital's rule be applied to solve this type of limit?
- Why does approximate to as ?
- What are some common pitfalls when solving trigonometric limits?
Tip: When solving limits involving trigonometric functions, remember that small-angle approximations can simplify the calculations significantly, especially when approaches 0.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
Small-angle approximations
Formulas
lim (x -> 0) (tan(x) ≈ x)
lim (x -> 0) (1 - cos(x) ≈ x^2 / 2)
Theorems
Limit properties
Small-angle approximations
Suitable Grade Level
Grade 12 or early college
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