Math Problem Statement
Solution
To analyze the statements in this image and confirm which ones are correct, let's go through each option in detail. Here’s a breakdown of each statement in the image:
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Statement 1:
This statement talks about two functions and defined in some neighborhood of point with limits: It then mentions that if , then and are infinitesimal functions of the same order of smallness as .
This is a true statement. When two functions both tend to zero and their ratio tends to a non-zero constant, they are of the same order of infinitesimality as . -
Statement 2:
This statement assumes is an infinitesimally small quantity as (where ). It then claims: This is also true. For small values of , , as they behave similarly in terms of asymptotics. -
Statement 3:
This statement considers two functions and in a neighborhood of point , with: It states that if both functions have limits as , then .
This is indeed correct and aligns with the properties of limits. If in some neighborhood of , then their limits must satisfy . -
Statement 4:
This statement assumes is an infinitesimally small quantity as . It claims: This is also correct. For small , due to the first-order Taylor expansion of the logarithm function.
Conclusion
Based on the detailed analysis, all four statements appear to be correct. Since the last statement is already selected in the image, it suggests that this answer was correctly identified.
Would you like further details on any of these points, or do you have any specific questions?
Related Questions
- What does it mean for two functions to be of "the same order of smallness" as ?
- How is the asymptotic equivalence between and derived for small ?
- How does the comparison theorem for limits apply in cases with inequalities between functions?
- Why is the approximation valid for small ?
- Can you give examples of functions that exhibit similar behavior to the ones in these statements?
Tip
When working with asymptotics, Taylor series expansions often provide powerful approximations for functions near specific points.
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Math Problem Analysis
Mathematical Concepts
Infinitesimal Analysis
Asymptotic Behavior
Limits
Taylor Expansion
Formulas
lim(x→a) f(x) = 0
lim(x→a) g(x) = 0
lim(x→a) (f(x)/g(x)) = C ≠ 0
sin α ≈ arcsin α for small α
log_a(1 + α) ≈ α / ln(a) for small α
Theorems
Limit Comparison Theorem
Taylor Series Expansion
Suitable Grade Level
Undergraduate Mathematics
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