Math Problem Statement
Solution
It appears that you've uploaded an image containing a series of mathematical questions. From the content visible, the questions seem to involve limits, asymptotic behavior of functions, and analysis around specific points (like ).
I will transcribe and analyze the key parts from the image:
Câu 1.19:
This is a standard limit result in calculus. The answer is:
Thus, the correct answer is .
Câu 1.20:
The behavior of as suggests analyzing the asymptotic expansion of , so the limit simplifies to something of the form:
So, the limit tends towards infinity as , which means the correct answer here is .
Câu 1.21:
Finding the limit:
For small , we can expand , and after substitution, the leading terms cancel out, simplifying to a more straightforward analysis of the limit. This leads to the result that the limit tends towards , so the correct answer is .
Let me know if you'd like further clarification or help with any of these steps.
Follow-up Questions:
- How do you evaluate limits involving exponentials as ?
- What is the asymptotic expansion of around ?
- How can L'Hopital's Rule be used to solve indeterminate limits?
- What are the standard trigonometric limits near ?
- How does the behavior of influence the limit of the given function as ?
Tip:
When dealing with limits involving small , asymptotic expansions can simplify complex expressions dramatically, leading to easier computation of the limit.
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Math Problem Analysis
Mathematical Concepts
Limits
Asymptotic Behavior
Exponential Functions
Trigonometric Limits
Formulas
lim (x -> 0) sin(x)/x = 1
Asymptotic expansion of e^x - 1 for small x
Theorems
L'Hopital's Rule
Standard trigonometric limits as x approaches 0
Suitable Grade Level
High School to College level (Grades 10-12, Calculus level)
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