Definição epsilon delta

The Precise Definition of Limit: δ-Proofs Math 109

Dr. Evans Spring 2007

The precise definition of limit given on page 50 in section 1.8 was developed over hundreds of years. It is therefore no surprise that students often have difficulties quickly mastering the definition and how to use it. The following remarks and examples are meant to serve as a guide as you learn to use use the definition to write your own δ-proofs. Please be aware as you study this that styles in proof writing vary greatly, and no two people will ever write the exact same proof. However, the logical structure of any two proofs (of the same result) should be more or less the same. In the present setting, you must demonstrate, usually through a series of algebraic steps, that the implication in the definition holds. In an δ-proof, you must first do some calculations to find the number δ, but these calculations are not part of the proof. Instead, the proof consists of specifying a value for δ in terms of and showing that the implication in the limit definition holds for this value of δ. Let’s begin by making a few remarks about absolute value. Absolute Value. First recall that if x is a real number, the absolute value of x is the distance from x to 0 and is written |x|. Said another way, we can define |x| = Therefore, if c is any real number, we have |x − c| = x − c if x ≥ c; c − x if x < c x if x ≥ 0; −x if x < 0.

so that it is natural (and useful) to think of |x − c| as the distance from x to c. Two important equivalences involving absolute value are |x − c| < δ ⇐⇒ −δ < x − c < δ ⇐⇒ c − δ < x < c + δ where the symbol ⇐⇒ means “is equivalent to”. In words, these equivalences say that x is less than δ units from c if and only if the difference x − c is between −δ and δ if and only if x is in the interval (c − δ, c + δ). Draw a picture! The Definition. Let us state the definition of limit, first informally and then precisely. Definition (informal). If f (x) is a function defined for all values of x near