What Is a Limit of a Function? A 5 Minute Introduction

[Greg Bernhardt]

Definition/Summary
Limits are a mathematical tool that is used to define the ‘limiting value’ of a function i.e. the value a function seems to approach when its argument(s) approach a particular value. Although the argument of the function can be taken to approach any value, limits are helpful in cases where the argument approaches a value where the function is not defined or becomes exceedingly large.
While defining a limit, we say that the argument ‘tends to’ a value. For example,
$$\lim_{x \to c}f(x) = m$$
is said: “As x tends to c, the function f(x) tends to m”. This statement however makes no assertion of what the value of f(c) would be. Rather, it means that as x becomes exceedingly close to ‘c’, f(x) becomes exceedingly close to m.
If, however, the function is defined and continuous at ‘c’, then:
$$\lim_{x \to c}f(x) = f(c)$$
(See explanation)
Equations
Identities of limits:
$$\lim_{x \to c}f(x) + g(x) = \lim_{x \to c}f(x) + \lim_{x \to c}g(x)$$...

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[fresh_42]

This has not to be confused with limits of sequences of functions as considered in functional analysis, e.g. example 5.9. in https://www.physicsforums.com/insights/hilbert-spaces-relatives/

 

[guhag]

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.