
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Using the property that in finding the limit of the ratio of two infinitesimals, they can be replaced with their equivalent infinitesimals (property II), find the following limits:
1) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\)
2) \(\lim _{x \rightarrow 0} \frac{\tan^{2} 2 x}{\sin ^{2} \frac{x}{3}}\)
3) \(\lim _{x \rightarrow 0} \frac{x \sin 2 x}{(\arctan 5 x)^{2}}\)
4) \(\lim _{n \rightarrow+\infty} \frac{\tan^{3} \frac{1}{n} \cdot \arctan \frac{3}{n \sqrt{n}}}{\frac{2}{n^{3}} \cdot \tan \frac{1}{\sqrt{n}} \cdot \arcsin \frac{5}{n}}\)

Present the answer in LaTex format: \boxed{Your answer}
user