2 LIMITS

SECTION 2.4THE PRECISE DEFINITION OF A LIMITWORKSHOP/DISCUSSION• Estimate how close x must be to 0 to ensure that (sin x) /x is within 0.03 of 1. Then estimate how close xmustbeto0toensurethat(sin x) /x is within 0.001 of 1. Describe what you did in terms of the definitionof a limit.• Discuss why f (x) = [[x]] does not have a limit atx = 0, first using the “narrow range” definition oflimit, and then possibly the ε-δ definition of limit.x 2 + xThen discuss why lim = 1, using thex→0 x“narrow range” definition of limit and a graph likethe one at right.y21_1 01 x• Find, numerically or algebraically, a δ > 0 such that if 0 ≤ |x − 0| ≤ δ, then ∣ ∣x 3 − 0 ∣ < 10 −3 . Similarly,compute a δ > 0suchthatif0≤ |x − 2| ≤ δ, then ∣ ∣x 3 − 8 ∣ < 10 −3 .GROUP WORK 1: A “Jittery” FunctionThis exercise can be done several ways. After they have worked for a while, perhaps ask one group to tryto solve it using the Squeeze Theorem, another to solve it using the “narrow range” definition of limit, and athird to solve it using the ε-δ definition of limit. They should show why their method works for Problem 2,and fails for Problem 3.Answers:1._1y10 1 x2. lim f (x) = 0. Choose ε with ε > 0. Let δ = √ ε.Nowx→0if −δ < x < δ, thenx 2 < ε, regardless of whether xis rational or irrational. This can also be shown using theSqueeze Theorem and the fact that 0 < f (x) < x 2 ,andthen using the Limit Laws to compute lim 0 and lim x 2 .x→0 x→03. It does not exist. Assume that lim f (x) = L. Choosex→1ε =10 1 . Now, whatever your choice of δ, there are somex-values in the interval (1 + δ, 1 − δ) with f (x) = 0, soL must be less than10 1 . But there are also values of x in the interval with f (x) > 10 2 ,soL must be greaterthan10 1 .SoL cannot exist. The “narrow range” definition of limit can also be used to solve this problem.4. We can conjecture that the limit does not exist by applying the reasoning from Problem 3.59