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# lec03 - 3 Formal Denition of a Limit 1 Lecture Notes 3...

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3. Formal Definition of a Limit 1 Lecture Notes: 3. Formal Definition of a Limit These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Formal Definition of a Limit Our definition of the limit lim x x 0 f ( x ) = L so far has been somewhat intuitive and not mathematically rigorous. We will look more closely at the formal definition of a limit in what follows. Definition: We say lim x x 0 f ( x ) = L if, for every ε > 0, there exists δ > 0 such that | f ( x ) - L | < ε whenever 0 < | x - x 0 | < δ. Let’s break down the pieces of the definition. Assume we’re given a number ε > 0, however small (and typically ε is thought to be a very small number). A requirement in the definition is that we have | f ( x ) - L | < ε or - ε < f ( x ) - L < ε i.e., L - ε < f ( x ) < L + ε. This means that the y -value of the curve y = f ( x ) must lie somewhere in the interval ( L - ε, L + ε ), as shown on the graph below.

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3. Formal Definition of a Limit 2 We will be picking a δ > 0 such that a couple of things happen. The first is that we’ll be dealing only with x -values such that 0 < | x - x 0 | < δ. (1) First, the part | x - x 0 | < δ means that - δ < x - x 0 < δ or x 0 - δ < x < x 0 + δ or that x ( x 0 - δ, x 0 + δ ) , i.e., x is constrained to remain in an interval around x 0 . The other part of the inequality in (1) is that 0 < | x - x 0 | . Since the absolute value is always a nonnegative value, the only way that we could violate this condition is if 0 = | x - x 0 | or x = x 0 . Thus the conditions given in the inequality in (1) give us that x ( x 0 - δ, x 0 + δ ) , x 6 = x 0 , or x ( x 0 - δ, x 0 ) ( x 0 , x 0 + δ ) . The selection of δ > 0 should then be such that if x ( x 0 - δ, x 0 + δ ) , x 6 = x 0 , then we are assured that f ( x ) ( L - ε, L + ε ) . Note that we are not at all concerned about the value of f ( x ) at x 0 , only the values of f ( x ) for x near x 0 . Also, as ε gets smaller, we will have to make δ smaller.
3. Formal Definition of a Limit 3 Note: It is clear that the choice of δ depends on ε ; however it also depends on the steepness of the curve y = f ( x ) and the location x 0 where the limit is being evaluated: Example 1.1 : In this example we will investigate the limit lim x 3 (2 x + 2) = 8 .

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lec03 - 3 Formal Denition of a Limit 1 Lecture Notes 3...

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