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Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Examples Example 1  
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Example 1
Consider the limit When x = 0, the function f(x) = 1 + 10x^{2}/x is undefined. When x is a real number close to but not equal to 0, f(x) is close to 1. Now let us be more explicit. How should we choose x to get f(x) strictly within 1/5 of 1? To solve this problem we assume x is strictly within some distance δ of 0 and get inequalities for f(x). By the lemma, we must find a δ > 0 such that whenever  δ < x < δ and x ≠ 0, we have 1  1/5 < f(x) < 1 + 1/5. Assume
Then 1  10δ < f(x) < 1 + 10δ. If we set δ = 1/50 , then 1  1/5 < f(x) < 1 + 1/5 This shows that whenever 1/50 < x < 1/50 and x ≠ 0, 1  1/5 < f(x) < 1 + 1/5. In other words, whenever 0 < x < 1/50,  f(x)  1 < 1/5. A similar computation shows that for each ε > 0, if 0 < x < ε/10 then f(x)  1 < ε. Thus the ε, δ condition for lim_{x→0} (1 + 10x^{2}/x) = 1 is true, and, for a given ε, a corresponding δ is δ = ε/10.


Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Examples Example 1 