In summary, the conversation is about the concept of limits in mathematics and its importance in calculus. The definition of a limit and its role in defining other mathematical concepts is discussed. The conversation also touches on the use of limits in functional analysis and the desire to learn more about mathematical topics such as linear algebra, group theory, and analysis for personal satisfaction. The speaker also mentions struggling to connect the concepts of logic and truth tables to more advanced topics like delta-epsilon limits. They seek advice on how to approach learning these topics.
ToggleWhat is a limit?What is a function?
Definition of a Limit of  a FunctionEquationsExtended explanation
What is a limit?
In mathematics, a limit is a fundamental concept used to describe the behavior of a function or sequence as it approaches a particular point or value. Limits play a crucial role in calculus, where they are used to define concepts like continuity, derivatives, and integrals.
Here are key aspects of limits:

Definition: The limit of a function or...

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Hi Greg. I am trying to develop the skill of "proofs based reasoning". I am a synthetic organic chemist, and would love to learn the math associated with chemistry, for example: Linear algebra, Group theory (Quantum chem). I would also like to learn to write, analyze and appreciate proofs in Analysis, for instance to read and work out problems in Apostol, Spivak.

I don't have any agenda/timelines (except in this lifetime, that'd be good!) and just would like to do it for self-satisfaction. I tried reading "Intro to logic" , truth tables (If p then q,) but I fail to see the jump from those to, say, delta-epsilon limits proof.

Maybe this is all silly... but in case it is not, would you be able to suggest/point out how I could go about this? Thanks and appreciate your time and help.

## 1. What is a limit of a function?

A limit of a function is a fundamental concept in calculus that represents the value that a function approaches as its input (x) approaches a particular value (a). It is often used to analyze the behavior of a function near a certain point and can help determine the continuity and differentiability of a function.

## 2. How is a limit of a function calculated?

A limit of a function can be calculated using various methods, including direct substitution, algebraic manipulation, and graphical analysis. However, the most commonly used method is the epsilon-delta definition, which involves setting a small value (epsilon) and finding a corresponding small value (delta) that ensures the function's output (f(x)) stays within a certain range of the limit value.

## 3. What is the difference between a left-hand limit and a right-hand limit?

A left-hand limit is the value that a function approaches from the left side of a particular point, while a right-hand limit is the value that a function approaches from the right side of that same point. If these two values are equal, the overall limit of the function exists, and the function is said to be continuous at that point.

## 4. Can a function have a limit if it is not defined at a certain point?

Yes, a function can have a limit even if it is not defined at a particular point. This is because the limit only considers the behavior of the function near that point and not necessarily at that point itself. However, if the left-hand and right-hand limits do not equal each other, the overall limit does not exist, and the function is said to have a discontinuity at that point.

## 5. Why are limits important in mathematics and science?

Limits are important in mathematics and science because they provide a way to analyze and understand the behavior of functions. They are used in various applications, such as calculating derivatives in calculus, determining the stability of systems in physics, and analyzing data in statistics. Limits also play a crucial role in theoretical concepts, such as infinity and continuity, and help to define fundamental concepts in mathematics and science.

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