Does the value of m n change in this discussion? Call m n, a. What does x approach as a limit? What is a? (The sign means approach as a limit.) We write xa. Read the statement. Is a x a variable? What does it approach as a limit? Make the statement using the above symbol. 194. 1. Theorem. We now wish to discuss two variables, each approaching its respective limit. If the variables have a constant ratio, what can we say of their limits? 2. 3. Let a and a' be two constants, and x and x' two variables approaching the respective constants as limits. 4. Also let += the constant, r. 5. [Note the variable xa.] 7. Get a clear notion of the ratio, stant, although x and x are variables. The ratio is con 8. Those who fail to get a clear conception of the mean ing of the above should study the concrete illustration given below: For the purpose of illustration we take x, x', a, a', known lines. In the theorem x, x', a, a' are unknown quantities; it (4) No matter how closely x = a, and x'=a', 2 х α x In this illustration we easily see that =2, because x a and a' are known, being 6 inches and 4 inches, respectively. 9. In order to grasp the full force of this proof it is absolutely necessary for the pupil (1) To get a clear idea of the meaning of each term used, (2) And to hold that idea firmly in mind during the entire discussion, (3) And to understand fully the authority for each statement made, (4) And to keep constantly in mind the steps whereby each statement is established. a > or < r. [?] I. Prove is not greater than r. Suppose, if possible, that α [When the numerator is constant, the value of a fraction is decreased by increasing the denominator. Then let b represent the required increase.] (3) Also, a = (a + b) r. [?] (4) But x = rx'; [?] ... a x may be made as small as we choose. - (7) And (a' x') r + b r = (a' — x' + b) r. [?] (8) bra constant positive quantity. [?] (9) (a' — x') r = an indefinitely small but positive quan based upon (6), (7), (8), (9), (10), and (11), all of which are proved by Geometry.] a (13) Hence is not greater than r. [?] a Prove is not less than r. a (5) ... a x = (a' — x' —b) r. [?] (C) But ax is positive, [?] (7) And (a' -xb) r is negative; [?] [The final reasoning is left to the pupil.] 195. Cor. I. If, while approaching their respective limits, two variables are always equal, are their limits equal? Give the proof of your answer. 196. Cor. II. If, while approaching their respective limits, two variables have a constant ratio, and one of them is always greater than the other, what do you conclude about their respective limits? Prove your answer. PROPORTIONAL LINES. 1. When a line drawn parallel to the base of a bisects one side, how will it meet the other side? 2. When a line is drawn parallel to one of the parallel sides of a trapezoid and bisects one of the non-parallel sides, how does it meet the other non-parallel side? 3. When parallel lines cut off equal parts on a given transversal, how will they meet any other transversal? 4. Draw a scalene triangle, A B C, and a line m n to A B cutting off of A C; what part of B C is cut off? Let ACB be any A and M any point in A C, and MN to A B. Compare the ratio of the parts of A C(A M: M C) with the ratio of the parts of BC (BN: N C). Case I. Suppose that CM and M A are commensurable. [Review § 172.] Apply a common measure, k, to C M and M A. Then A M=pk, and M C = qk. [?] Through points of division of A M and MC draw lines parallel to A B. |