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LIMITS AND CONTINUITY OF FUNCTIONS Introduction to Limits “ I’m nearing the limit of my patience” Consider the function determined by the formula Note that is not defined at x = 1 since at this point f(x) has the form 0/0, which is meaningless. What is happening to f(x) as x approaches 1? Is f(x) approaching some specific number as x approaches 1?x approaches 1

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All the information we have assembled seems to point to the same conclusion: f(x) approaches 3 as x approaches 1. In mathematical symbols, we write This is read “ the limit as x approaches 1 of is 3.” (Intuitive meaning of limit) To say that means that when x is near but different from c, then f(x) is near L.

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Definition (Right- and Left-hand limits) To say that means that when x near but on the right of c, then f(x) is near L. Similarly, To say that means that when x near but on the left of c, then f(x) is near L.

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Example. Sketch the graph of then find each of following or state that it does not exist. (a) f(1) (b) (c) (d) Theorem if and only if and

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Definition (Precise meaning of limit) To say that means that for each given (no matter how small), there is a corresponding such that provided that ; That is,

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There is >0 such that For every >0

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