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Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Lemma and Definition for Limits  
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Lemma and Definition for Limits
LEMMA (i) x is within δ of c if and only if c  δ ≤ x ≤ c + δ. (ii) x is strictly within δ of c if and only if c  δ < x < c + δ. PROOF (i) Subtracting c from each term we see that c  δ ≤ x ≤ c + δ if and only if  δ ≤ x  c ≤ δ, which is true if and only if x  c ≤ δ. The proof of (ii) is similar. We shall repeat our infinitesimal definition of limit from Section 3.3 and then write down the ε, δ condition for limits. Later we shall prove that the two definitions of limit are equivalent to each other. Suppose the real function f is defined for all real numbers x ≠ c in some neighborhood of c. DEFINITION OF LIMIT (Repeated) The equation lim_{x→c} f(x) = L means that whenever a hyperreal number x is infinitely close to but not equal to c, f(x) is infinitely close to L. ε, δ CONDITION FOR lim_{x→c }f(x) = L For every real number ε > 0 there is a real number δ > 0 which depends on ε such that whenever x is strictly within δ of c but not equal to c, f(x) is strictly within ε of L. In symbols, if 0 < x  c < δ, then f(x)  L < ε. In the ε, δ condition, the notion of being infinitely close to c is replaced by being strictly within δ of c, and being infinitely close to L is replaced by being strictly within ε, of L. But why are there two numbers ε and δ, instead of just one? And why should δ depend on ε? Let us look at a simple example.


Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Lemma and Definition for Limits 