How to do question 33 EXAMPLE 4 Prove that lim x² = 9.SOLUTION1. Guessing a value for 8. Let ɛ > 0 be given. We have to find a number 8 >0such thatif0 < |x – 3|< 8then|x² – 9| < ɛTo connect |x? – 9| with |x – 3| we write |x? – 9| = |(x + 3)(x – 3)|. Thenwe want0 < |x – 3|< 8|x + 3||x – 3|< 8ifthenNotice that if we can find a positive constant C such that |x + 3| < C, then|x + 3||x – 3|< C[x – 3|and we can make C|x – 3|< e by taking |x – 3|< 8/C, so we could choose8 = ɛ/C.We can find such a number C if we restrict x to lie in some interval centered at 3.In fact, since we are interested only in values of x that are close to 3, it is reasonableto assume that x is within a distance 1 from 3, that is, x – 3 < 1. Then 2 < x < 4,so 5 0, let 8 = min{1, ɛ/7}. If0< |x – 3|< 8, then |x – 3|<1 → 2 33. Verify that another possible choice of ô for showing thatlimx-3 x? = 9 in Example 4 is ômin{2, ɛ/8}.=

Name: How to do question 33 EXAMPLE 4 Prove that lim x² = 9.SOLUTION1. Guess
Uploaded: 2024-03-05T11:52:21.000Z
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Description: How to do question 33 EXAMPLE 4 Prove that lim x² = 9.SOLUTION1. Guessing a value for 8. Let ɛ > 0 be given. We have to find a number 8 >0such thatif0 < |x – 3|< 8then|x² – 9| < ɛTo connect |x? – 9| with |x – 3| we write |x? – 9| = |(x + 3)(x – 3)|.