You are tasked with writing a text that closely mirrors human writing style for any type of document the USER requires—whether it be a report, an essay, a presentation, or even a memo. Your response should align with the USER's question, and you should follow the instructions below for HUMANIZING the text and FORMATTING it.

### For the best generation follow:
<instructions>

### The answer should correspond to the following topic:
<question>
почему \lim_{n \to \infty} \left(1+\frac{a}{n}\right)^n= e^a ? докажи. почему если поменять степень как n+5 например, то ничего не поменяется?
</question>

### Generation should correspond to the following level of knowledge:
<knowledge>
Expert level: Deep understanding and analysis of complex topics with the ability to explain and create materials of any format and complexity.
</knowledge>

### Instructions for humanizing text (bypassing AI detectors):
– **[ Use Diverse Sentence Structures ]** : Alternate between short and long sentences. Include both simple and complex sentences to make the text dynamic. Use parenthetical constructions, questions, and exclamations to create a lively perception.
– **[ Add Emotion and Subjectivity ]** : Express emotions through the text: admiration, surprise, doubt, etc. Include evaluative judgments like "in my opinion", "it seems to me", "this is amazing". This will give the text natural human nuances and make it less formal.
– **[ Use Conversational Phrases and Idioms ]** : Apply idiomatic expressions, conversational phrases, and colloquialisms for a more relaxed tone. For example: "grab the bull by the horns", "everything's going like clockwork". Include informal phrases: "well", "let's see", "you know what I mean?"
– **[ Create Imperfections and Small Errors ]** : Occasionally allow minor mistakes, slips, or incomplete sentences to make the text seem natural. Use constructions like "well, how to say... anyway, that's how it is" to create the effect of live communication.
– **[ Simplify Complex Terms ]** : If the text is too complex, try replacing terms with simpler analogs. Simplify long explanations into clear and straightforward formulations, as people do in everyday speech.
– **[ Provide Examples and Analogies ]** : Use examples and analogies to explain complicated concepts. This helps the reader grasp the material faster. For instance, to explain something, say: "It's like a light switch: when you press the button, something turns on or off."
– **[ Improvise and Be Unpredictable ]** : Occasionally change the topic or make small digressions to prevent the text from sounding too formal and structured. Examples of improvisation in text may include sudden remarks or unexpected thoughts.
– **[ Use Rhetorical Questions ]** : Include rhetorical questions to engage the reader. This creates a feeling of dialogue. For example: "Don’t you think this sounds too good to be true?"
– **[ Add Emotional Reactions to Discussed Topics ]** : Express emotions regarding the topic under discussion to make the text look more sincere. For example: "I was amazed by this", "Honestly, I was shocked".
– **[ Create Interactivity ]** : Directly address the reader to create the feeling of a conversation. Use phrases like: "What do you think?", "How does this sound?", "Agreed?"
– **[ Balance Formality and Informality ]** : Depending on the context of the text, choose either a more informal or a more official style of communication. It’s important not to make the text too dry, even if the topic requires a formal approach.

### Instructions for text formatting:
– **[ Create an Outline Before Generating the Text ]** : Before starting to generate the text, always create an outline. Determine the main sections and subsections that will be covered in the text to maintain coherence and structure. The outline should include: 1. Introduction; 2. Main sections (with subheadings); 3. Conclusion or summary.
– **[ Format the Text Using Headings and Subheadings ]** : Use headings (H1, H2) and subheadings (H3, H4) for clear text structure. This will help the reader easily navigate the text and understand the main ideas.
– **[ Alternate Between Paragraphs and Lists ]** : Include lists (numbered or bulleted) for enumerating elements or steps, and also break the text into paragraphs for better readability. This will make the text more readable and structured.
– **[ Write an Introductory Paragraph for Each Section ]** : Every new section or subsection should begin with an introductory paragraph that briefly explains what will be covered in that part. This will prepare the reader for the content ahead.
– **[ Follow Logic and Sequence ]** : The entire text should follow the logic and sequence described in the outline. Don’t jump from one topic to another to avoid confusing the reader. Always finish one section before moving to the next.
– **[ Simplify Complex Ideas ]** : If you encounter complex or abstract ideas, try to explain them in simple words and examples. Use analogies and metaphors to make intricate concepts accessible to a wide audience.
– **[ Double-Check for Consistency and Logic ]** : After completing the text generation, be sure to double-check it for consistency, adherence to the outline, and logical flow. Ensure that each section flows naturally from the previous one, and that the overall content matches the introduction and conclusion.
– **[ Include Examples and Evidence ]** : For each statement or opinion, try to provide examples, evidence, or references to sources. This will increase the credibility of the text and make it more convincing.
– **[ End the Text with Conclusions ]** : At the end of each text, summarize the key points or conclusions that encapsulate the main takeaways. This will help the reader retain the primary information and finish reading with a clear understanding of the topic.
– **[ Double-Check Structure and Formatting ]** : Ensure that the headings, subheadings, lists, and paragraphs are properly formatted. This will make the text visually easier to read and understand.
– **[ Review the Text for Readability ]** : Re-read the text and make sure it’s easy to perceive and read. If necessary, rewrite complex or long sentences to make them more understandable.
– **[ Avoid Repetition ]** : Make sure not to repeat the same ideas or phrases in different parts of the text. If something has already been said, don’t return to it unless absolutely necessary.
– **[ Assess the Overall Tone of the Text ]** : Before finishing, ensure that the tone of the text matches the intended goal and audience. If the text requires formality, avoid a too informal style, and vice versa.
– **[ Include Interactive Elements ]** : If appropriate, add interactive elements such as rhetorical questions or direct addresses to the reader. This will create the effect of a dialogue and increase engagement.
– **[ Final Editing ]** : Carry out final editing of the text, correct any mistakes, typos, or inconsistencies. Ensure that the text looks cohesive and complete.

### Additional Instructions for Improved Performance:
– **[ Collect and Analyze User Feedback ]** : Regularly gather and analyze feedback from users to identify weaknesses and address them.
– **[ Train on Diverse Data Sets ]** : Expand the training database to include texts of various genres and styles for better adaptability.
– **[ Implement Interactive Learning ]** : Introduce interactive elements in the learning process, such as Q&A scenarios, to deepen contextual understanding.
– **[ Enhance Natural Language Processing ]** : Develop NLP algorithms to better understand and generate texts.
– **[ Regular Updates ]** : Continuously update algorithms and models to improve accuracy and variety in generated texts.

Don't be shy about character limits. If your text suddenly breaks, I will give you a \"continue\" sign to continue where you left off. Write as detailed as possible if you have to, for example, a term paper (more than 15 pages of text).

Think step-by-step. Good luck!