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Sizes of Infinity 179 In general, the value of that will enable us to complete the proof depends on the behavior of the function around c, and smaller values of are more likely to work. In Example 4.5.15 we used = 1/2, but, if one looks carefully through the steps of the proof, any value of smaller than or equal to 1 (which is one of the values of the function) would be acceptable. Thus, in general the choice of is not unique. The proof techniques illustrated in the examples are not always easy to implement. As one advances in the study of real analysis, one builds more and more tools to deal efficiently with the nonexistence of limits. Some of these tools rely on the structure of the real numbers (e.g., density properties of rational and irrational numbers), and on the relationships between functions and sequences. Exercises 15. Prove that lim x3 ð2x + 2Þ = 8: 16. Prove that lim x1 ð3x 2 + 2Þ = 5: Check the result obtained for . 17. In Example 4.5.11, rework the proof by modifying the statement "We can arbitrarily limit the calcu- lation to values of x that are less than ½ unit from -2" to read "We can arbitrarily limit the calcula- tion to values of x that are less than 1 unit from 2." What expression does one obtain for ? 18. Prove that lim x2 x 2 1 1 = 1 : Check the result obtained for . 5 + 3 19. Prove that lim x1 x 2 - 1 = 3 : Check the result obtained for . 2 x - 1 20. The choice of is not unique. In the discussion following Example 4.5.10, we proved that when = 4.5, we can use = 1.5. Show that if we choose = 0.9, it is still true that |(3x - 5) - 1| < 4.5.