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 $A$.

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 $D$.

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 $D$.

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.