Math Problem Statement
Solution
The image contains a series of limit problems that involve trigonometric functions as . Let’s solve these limits one by one.
1.
Using the approximation as , we get: So, the value is 4.
2.
We use the approximations and as : So, the value is 6.
3.
Using and as : So, the value is .
4.
This is a standard limit that can be directly computed using the approximation as : So, the value is 1.
5.
Using as : So, the value is .
Would you like any further details or explanations on these solutions?
Here are some related questions to consider:
- How do approximations like help in limit problems?
- What is the general approach to solving limits involving trigonometric functions?
- Can you explain why is important?
- What happens if the limits involve higher powers of or trigonometric terms?
- How would the limits change if were approaching values other than 0?
Tip: Always remember that trigonometric approximations and are very useful for limit problems as .
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
Calculus
Formulas
lim (x -> 0) (tan^2(2x) / 2x^2) = 4
lim (x -> 0) (sin(6x) / (x * cos(2x))) = 6
lim (x -> 0) (sin(2x) / tan(8x)) = 1/4
lim (x -> 0) (x / sin(x)) = 1
lim (x -> 0) (tan(3x) / 5x) = 3/5
Theorems
Squeeze Theorem
Small-angle approximations: sin(x) ≈ x, tan(x) ≈ x as x -> 0
Suitable Grade Level
Grades 11-12
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