Schaum's Outline of Theory and Problems of Beginning Calculus

26 STRAIGHT LINES [CHAP. 4f’Fig. 4-3(4Now let us see how the slope varies with the “steepness” of the line. First let us consider lines withpositive slopes, passing through a fixed point P,(x,, yJ. One such line is shown in Fig. 4-4. Takeanother point, - P2(x2, y,), on A? such that x2 - x1 = 1. Then, - by definition, the slope rn is equal to thedistance RP, . Now as the steepness of the line increases, RP, increases without limit [see Fig. 4-5(a)J.Thus, the slope of 9 increases from 0 (when 9 is horizontal) to + CO (when 9 is vertical). By a similarconstruction we can show that as a negatively sloped line becomes steeper and steeper, the slope steadilydecreases from 0 (when the line is horizontal) to - CQ (when the line is vertical) [see Fig. 4-5(b)].t’ YFig. 44