LIMITS AND CONTINUITY OF FUNCTIONS Introduction to Limits “ I’m nearing the limit of my patience” Consider the function determined by the formula Note.

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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

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.

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.

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

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,

There is  >0 such that For every  >0

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