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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Sullivan 90 Chapter 1 • Limits and Continuity (b) { w|0< w ≤13} (c) C(w) 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 w (d) lim Cw ( ) = 0.94; lim Cw ( ) = 1.15; w →1 − w → 1 + lim Cw ( ) does not exist. w →1 (e) lim Cw ( ) = 3.25; w →12 − lim Cw ( ) = 3.46; lim Cw ( ) does + w → 12 w →12 not exist. (f) lim Cw ( ) = 0.94 w → 0 + (g) lim Cw ( ) = 3.46 − w →13 61. (a) lim St () = 100 t→7 (b) Given any ε > 0, there is a number δ > 0 such that whenever 0 < | t − 7| < δ. | St ( − 100)|
Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Section 1.2 • Limits of Functions Using Properties of Limits 91 y A f (x) 5 A THEOREM The Limit of a Constant If f (x) = A, where A is a constant, then for any real number c, lim f (x) = lim A = A (1) x→c x→c The theorem is proved in Section 1.6. See Figure 17 and Table 8 for graphical and numerical support of lim x→c A = A. Teaching Tip Depending on the function given, a graphical approach may be helpful. Consider verifying the limits using graphical and analytical (that is, algebraic) methods. Figure 17 For x close to c, the value of f remains at A; lim x→c A = A. y c c c f (x) 5 x Figure 18 For x close to c, the value of f is just as close to c; lim x→c x = c. x x TABLE 8 x approaches c from the left −−−−−−−−−−−−−−−−−−−−→ x approaches c from the right ←−−−−−−−−−−−−−−−−−−−−− x c− 0.01 c − 0.001 c − 0.0001 → c ← c + 0.0001 c + 0.001 c + 0.01 f (x) = A A A A f(x) remains at A A A A For example, lim x→5 2 = 2 lim 1 3 = 1 3 x→ √ 2 THEOREM The Limit of the Identity Function If f (x) = x, then for any number c, lim(−π) =−π x→5 lim f (x) = lim x = c (2) x→c x→c This theorem is proved in Section 1.6. See Figure 18 and Table 9 for graphical and numerical support of lim x→c x = c. TABLE 9 x approaches c from the left −−−−−−−−−−−−−−−−−−−−→ x approaches c from the right ←−−−−−−−−−−−−−−−−−−−−− x c− 0.01 c − 0.001 c − 0.0001 → c ← c + 0.0001 c + 0.001 c + 0.01 f (x) = x c− 0.01 c − 0.001 c − 0.0001 f (x) approaches cc+ 0.0001 c + 0.001 c + 0.01 For example, lim x→−5 x =−5 lim x→ √ x = √ 3 3 lim x = 0 x→0 1 Find the Limit of a Sum, a Difference, and a Product Many functions are combinations of sums, differences, and products of a constant function and the identity function. The following properties can be used to find the limit of such functions. Teaching Tip Students do not have to be able to state the theorems presented in this section. However, they must be able to apply them. Consider presenting these general rules: • When finding the limit of a function, first try to substitute the value of interest. If that results in a finite value, or something in the undefined form N for a nonzero N 0 (in which case the limit doesn’t exist), then you’re done. • If the substitution results in an indeterminate expression (like 0 0 or ∞ ∞ ), try to simplify the expression. If the expression is rational, try to factor the numerator and denominator and simplify. If the rational expression contains a binomial and one of the terms is a radical, try multiplying by the conjugate. • Remember, the ultimate goal is to determine what value the function is approaching as x gets very close to the given value of interest. IN WORDS The limit of the sum of two functions equals the sum of their limits. THEOREM Limit of a Sum If f and g are functions for which lim f (x) and lim g(x) both exist, x→c x→c then lim[ f (x) + g(x)] exists and x→c A proof is given in Appendix B. lim[ f (x) + g(x)] = lim f (x) + lim g(x) x→c x→c x→c TRM Section 1.2: Worksheet 1 This worksheet contains 6 questions for finding limits by direct substitution. Graphs are provided for answer verification. Teaching Tip As you get into the meat of this chapter, it may be time to refer to the pacing guide and adjust pacing. There are two schools of thought to consider: (1) teaching to mastery level initially, leaving only 3 weeks to review for the end-of-term exam, or (2) moving more quickly through the material, which then will leave 4 to 5 weeks for review. The rationale for the second approach is that many of the topics that students found difficult at the beginning of the fall semester are ones with which they are very comfortable by the midterm in December. Here in Chapter 1, they still don’t yet have the background and experience to ask the good questions needed to push their understanding of the concepts because they are still focused on the nuts and bolts of getting the correct answer. Section 1.2 • Limits of Functions Using Properties of Limits 91 TE_Sullivan_Chapter01_PART 0.indd 20 11/01/17 9:52 am
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