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Unformatted text preview: 2.4 The Precise Definition of a Limit Math 1271, TA: Amy DeCelles 1. Overview Definition of a Limit We say the limit of f ( x ) as x approaches a is L if the following condition is satisfied: For every number > 0 there is a number > 0 such that: if | x- a | < then | f ( x )- L | < Parsing this definition Our intuitive understanding is that L is the limit (of f ( x ) as x a ) if f ( x ) gets closer and closer to L as x gets closer and closer to a . Lets see how this matches up with the precise definition. First look at the expression | x- a | . This is the distance between x and a . So if we make smaller and smaller, that means that x is getting closer and closer to a . Similarly, | f ( x )- L | is the distance between the y-values f ( x ) and L , so if gets smaller and smaller, that means that f ( x ) is getting closer and closer to L . So, just to make this clear: is a distance that specifies an x-range: how far away from a can x be? ... it must be within units of...
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