Editing Limit of a function (section)

==Motivation==
Imagine a person walking on a landscape represented by the graph {{math|1=''y'' = ''f''(''x'')}}. Their horizontal position is given by {{mvar|x}}, much like the position given by a map of the land or by a [[Global Positioning System|global positioning system]]. Their altitude is given by the coordinate {{mvar|y}}. Suppose they walk towards a position {{math|1=''x'' = ''p''}}, as they get closer and closer to this point, they will notice that their altitude approaches a specific value {{mvar|L}}.  If asked about the altitude corresponding to {{math|1=''x'' = ''p''}}, they would reply by saying {{math|1=''y'' = ''L''}}.

What, then, does it mean to say, their altitude is approaching {{mvar|L}}? It means that their altitude gets nearer and nearer to {{mvar|L}}—except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of {{mvar|L}}. They report back that indeed, they can get within ten vertical meters of {{mvar|L}}, arguing that as long as they are within fifty horizontal meters of {{mvar|p}}, their altitude is ''always'' within ten meters of {{mvar|L}}.

The accuracy goal is then changed: can they get within one vertical meter? Yes, supposing that they are able to move within five horizontal meters of {{mvar|p}}, their altitude will always remain within one meter from the target altitude {{mvar|L}}. Summarizing the aforementioned concept we can say that the traveler's altitude approaches {{mvar|L}} as their horizontal position approaches {{mvar|p}}, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of {{mvar|p}} where all (not just some) altitudes correspond to all the horizontal positions, except maybe the horizontal position {{mvar|p}} itself, in that neighbourhood fulfill that accuracy goal.

The initial informal statement can now be explicated:

{{block indent|The limit of a function {{math|''f''(''x'')}} as {{mvar|x}} approaches {{mvar|p}} is a number {{mvar|L}} with the following property:  given any target distance from {{mvar|L}}, there is a distance from {{mvar|p}} within which the values of {{math|''f''(''x'')}} remain within the target distance.}}

In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a [[Hausdorff space|topological space]].

More specifically, to say that

<math display=block> \lim_{x \to p}f(x) = L,</math>

is to say that {{math|''f''(''x'')}} can be made as close to {{mvar|L}} as desired, by making {{mvar|x}} close enough, but not equal, to&nbsp;{{mvar|p}}.

The following definitions, known as {{math|(''ε'', ''δ'')}}-definitions, are the generally accepted definitions for the limit of a function in various contexts.