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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 δ > 0 such that: if  x a  < δ 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 . Let’s 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 δ smaller and smaller, that means that x is getting closer and closer to a . Similarly,  f ( x ) L  is the distance between the yvalues 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: δ is a distance that specifies an xrange: how far away from a can x be? ... it must be within δ units of...
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 Spring '08
 SPEZIALE
 Math, Calculus, Writing, Universal quantification, X, Amy DeCelles

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