My English lesson Horizontal asymptotes A horizontal asymptote is a line parallel to the x-axis to which the graph of the function progressively approaches: we can say that the straight-line is tangent to the graph of the function asymptotically. DEFINITION The straight-line y = l is a horizontal asymptote for a function y = f(x) if: lim f(x) = l x example x+4 O Verify, using the appropriate limits, that the function y = ____ has the straightx 2 line x = 2 as its vertical asymptote and has the straight-line y = 1 as its horizontal asymptote. We know that the graph of the function is an equilateral hyperbola whose centre is in C(2 ; 1) and that the straight-line y = 1 is a horizontal asymptote (figure below). Anyhow, let us verify by using the definition that lim f(x) = 1. BE CAREFUL! B R Remember that a function having ax + b equation y = ______ with c 0 cx + d and a d = b c has a vertical c asymptote of equation x = __ and d a y = __ as its horizontal asymptote c (see unit 1, §3). x y 1 4 2 4 2 O 2 2 4 x We shall verify that for each neighborhood I1; centered in 1 it exists a neighborhood of infinity I so that: x I f(x) I1; in other words, if a real positive number is chosen arbitrarily, it exists a real positive number M so that if |x| > M then f(x) belongs to the neighborhood of 1 of ray : |x| > M |f(x) 1| 0 a k R can be determined so that if x > k then |f(x) l | 0 a k R can be determined so that if x 1 ____ __ 6 |x 2| > __ 6 6 From which: x 2 + __ ; since > 0 is the ray of the neighborhood of 6 6 1, it means that 2 __ 0. By giving the value M to the bigger one amongst them, we have determined the neighborhood of the infinity I we were looking for that verifies the limit. The straight-line y = 1 is a horizontal asymptote for the function. NOW IT S YOUR TURN N 4 Given the function y = __2 verify x that the straight-line y = 0 is a horizontal asymptote for the function. 127