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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 \( x \to 0 \)).

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

### Câu 1.19:
\[ \lim_{x \to 0} \frac{\sin x}{x} \]
This is a standard limit result in calculus. The answer is:

\[
\lim_{x \to 0} \frac{\sin x}{x} = 1
\]

Thus, the correct answer is \( A \).

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### Câu 1.20:
\[ \lim_{x \to 0} f(x) \sim \frac{e^x - 1}{x^2} \]

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

\[
f(x) \sim \frac{x}{x^2} = \frac{1}{x}
\]

So, the limit tends towards infinity as \( x \to 0 \), which means the correct answer here is \( D \).

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### Câu 1.21:
Finding the limit:
\[ \lim_{x \to 0} f(x) = \frac{e^x - 1 - \frac{x^2}{2}}{2x + x^2} \]

For small \( x \), we can expand \( e^x - 1 \approx x + \frac{x^2}{2} \), 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 \( 0 \), so the correct answer is \( D \).

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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 \( x \to 0 \)?
2. What is the asymptotic expansion of \( e^x \) around \( x = 0 \)?
3. How can L'Hopital's Rule be used to solve indeterminate limits?
4. What are the standard trigonometric limits near \( x = 0 \)?
5. How does the behavior of \( \cos x \) influence the limit of the given function as \( x \to 0 \)?

#### Tip:
When dealing with limits involving small \( x \), asymptotic expansions can simplify complex expressions dramatically, leading to easier computation of the limit.

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.

This problem involves evaluating limits such as lim(x -> 0) sin(x)/x and analyzing the asymptotic behavior of exponential functions. Explore detailed step-by-step solutions using key calculus concepts and theorems such as L'Hopital's Rule and trigonometric limits.

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