Math Problem Statement

Evaluate the limits and analyze the asymptotic behavior of functions as x approaches 0. Questions involve standard limit problems, such as sin(x)/x and functions involving exponential terms.

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 x0x \to 0).

I will transcribe and analyze the key parts from the image:

Câu 1.19:

limx0sinxx\lim_{x \to 0} \frac{\sin x}{x} This is a standard limit result in calculus. The answer is:

limx0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1

Thus, the correct answer is AA.


Câu 1.20:

limx0f(x)ex1x2\lim_{x \to 0} f(x) \sim \frac{e^x - 1}{x^2}

The behavior of f(x)f(x) as x0x \to 0 suggests analyzing the asymptotic expansion of ex1xe^x - 1 \approx x, so the limit simplifies to something of the form:

f(x)xx2=1xf(x) \sim \frac{x}{x^2} = \frac{1}{x}

So, the limit tends towards infinity as x0x \to 0, which means the correct answer here is DD.


Câu 1.21:

Finding the limit: limx0f(x)=ex1x222x+x2\lim_{x \to 0} f(x) = \frac{e^x - 1 - \frac{x^2}{2}}{2x + x^2}

For small xx, we can expand ex1x+x22e^x - 1 \approx x + \frac{x^2}{2}, 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 00, so the correct answer is DD.


Let me know if you'd like further clarification or help with any of these steps.

Follow-up Questions:

  1. How do you evaluate limits involving exponentials as x0x \to 0?
  2. What is the asymptotic expansion of exe^x around x=0x = 0?
  3. How can L'Hopital's Rule be used to solve indeterminate limits?
  4. What are the standard trigonometric limits near x=0x = 0?
  5. How does the behavior of cosx\cos x influence the limit of the given function as x0x \to 0?

Tip:

When dealing with limits involving small xx, 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)