Math Problem Statement
Solution
We are tasked with evaluating the following limit:
Let's solve this step by step.
Step 1: Apply small-angle approximations
For small , we can use the following approximations for and :
- when is close to 0.
- when is close to 0.
Thus:
- for small ,
- , and therefore .
Step 2: Substitute these approximations
Substitute these small-angle approximations into the given expression:
Thus, the limit becomes:
Step 3: Simplify the expression
Simplifying the above expression:
Final Answer:
If you have any questions or want more details, feel free to ask!
Related Questions:
- How do you apply L'Hôpital's Rule to limits involving indeterminate forms?
- What are the common small-angle approximations for trigonometric functions?
- How can you determine if a limit results in an indeterminate form?
- What happens if the angle in or isn't small enough to use the approximation?
- What other techniques can be used for evaluating limits as ?
Tip:
Always remember that small-angle approximations like and are valid when is close to 0, and can significantly simplify limit problems.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
Small-Angle Approximation
Formulas
tan(x) ≈ x for small x
sin(x) ≈ x for small x
Theorems
Small-Angle Approximation Theorem
Suitable Grade Level
Grades 10-12 or college-level calculus
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