Analytical definitions vs intuitive (or perhaps first year ) definitions

In summary, there are distinct differences between analytical definitions and intuitive or first year definitions, with first year books tending to use implications rather than precise definitions. The limit of a function at infinity, as defined in a first year calculus book, is not as precise as the definition found in a real analysis textbook, which uses the epsilon-delta definition.
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Analytical definitions vs intuitive (or perhaps "first year") definitions

I just began my real analysis course in college and we were given an assignment; a bunch of mathematical terms for us to define. We are asked to define them using two textbooks, one, our first year calculus textbook, the other, our real analysis textbook. The prof noted that there will be distinct differences between the two. He said "first year calculus books tend to use 'implications' as primary definitions, rather than precise definitions."

My problem is, the analysis textbook is very expensive and my current financial situation has not allowed for me to obtain the text immediately. So I can't look up anything in the analysis text.

My question is that I am unsure of what to look for...here, I'll give you an example.

We are asked to define the limit of a function at infinity: Let f be a function defined on some interval (a, ∞). Then lim┬(x→∞) f(x) = L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. For negatives, let f be a function defined on some interval (–∞, a). Then lim┬(x→-∞) f(x)=L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.

This is my definition from the first year book (stewart) that I chose. Now, this doesn't seem very "implicative" as my prof noted; to the contrary, it seems rather precise. Is the aforementioned definition the one that I would call the "analysis" definition? Or is this the "first year" definition? If this is simply the "first year" definition, then I am blown away by how much more math I have to learn (then again, either way, I know how much more there is to learn).

The way I see it, the above definition I gave seems to be the "real analysis" definition, and a simpler, "first year" definition would be something like: the limit of a function at infinity is finding the value of the limit of a function as its input becomes infinitely large.

I hope my question is clear, thank you all in advance and I apologize if this should have been posted elsewhere (eg homework forum).
 
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I don't really know what your prof means by first-year texts using "'implications' rather than precise definitions", as if implications are vague in general.

Anyway, I think this Stewart definition qualifies as a "first year" definition. It is not very precise because what do "arbitrarily close" and "sufficiently large" mean exactly? If you look at a real analysis book you'll see how to make this precise: the epsilon-delta definition.
 

What is the difference between analytical definitions and intuitive (or first year) definitions?

Analytical definitions are precise and specific, using logical and mathematical methods to define a concept. Intuitive or first year definitions are more general and based on personal understanding or common sense.

Which type of definition is more commonly used in scientific research?

Analytical definitions are more commonly used in scientific research as they provide a clear and unambiguous understanding of a concept.

Why are intuitive definitions not as reliable as analytical definitions?

Intuitive definitions can vary from person to person and can be influenced by personal biases and experiences. This makes them less reliable compared to analytical definitions, which are based on logical and mathematical principles.

Can intuitive definitions be used in scientific research?

Intuitive definitions can be used in scientific research as a starting point, but they may need to be refined or replaced with analytical definitions for more accurate and precise results.

How do analytical definitions and intuitive definitions complement each other?

Analytical definitions provide a solid foundation for understanding a concept, while intuitive definitions can help researchers think creatively and come up with new ideas and hypotheses.

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