Explain the key concepts and calculations shown regarding asymptotes and limits

ASYMPTOTES In Chapter 2 we studied rational functions, and saw how they are characterised by the presence of asymptotes. Consider the function f(x) = 2x + 3 which has domain x - 4 YA *=4 \\ f(@) = 21 + 3 {x x * 4, r ER). x-4 From the graph of y = f(x), we can see the function has 2 a vertical asymptote x = 4, and a horizontal asymptote Nico 14 y = 2. Both of these asymptotes can be described in terms of limits: As x-47, f (20 ) - -00 x -+ 4 reads As x- 47, f (20) - +00 "ac tends to 4 from the left". As x - -00, f (20) - 2- f (a) - 2 reads As * - too, f(2) - 2+ " f (z) tends to 2 from Since f(x) converges to a finite value as x - -00, below" we see lim f(x) = 2. I- -90 Since f(x) converges to a finite value as x - too, we see lim f(x) = 2. I-+00 This matches what we see algebraically, since lim 2x + 3 lim 20 + 3 and 1-00 * - 4 2+- 3 lim lim x- -00 1 - = 2 = 2