My English lesson

My English lesson Horizontal and vertical asymptotes Vertical asymptotes A vertical asymptote is a line parallel to the y-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. Hence, the graph of the function y = f(x) has as its vertical asymptote the straightline x = a if the following statements are true: Q the function is not defined for x = a; Q lim f(x) = . x a DEFINITION The line x = a is a vertical asymptote for a function y = f(x) if both of the following conditions are verified: Q the function is not defined for x = a; Q lim f(x) = x a y 3 2 1 O 3 2 1 1 It is worth considering that if a function is not defined in a point, not necessarily its graph has a vertical asymptote in that point. 1 2 3 x 2 3 x2 For example, the function y = __ is not defined only for x = 0 and its graph is the x bisector of quadrants I and III (side figure) without the point (0 ; 0) marked by the empty point on O. As you may see in the graph, the function does not have any vertical asymptote. example 3 O Verify that the function y = _____ has the line x = 2 as its vertical asymptote. x 2 3 The function y = ____ is defined for x R, x 2 (the denominator equals zero for x 2 x = 2). We know that the graph of the function is an equilateral hyperbola whose centre is in C(2 ; 0) and that the straight-line x = 2 is a vertical asymptote (figure below). Anyhow, let us verify by using the definition that lim f(x) = . y x 2 As for the definition of limit, we shall verify that for each neighborhood of infinity J it exists a neighborhood I2; centered in 2 so that x I2; (with x 2) f(x) J . In other words, if a real positive number M is chosen arbitrarily, it exists a real positive number such as for each x (different from 2) that belongs to the neighborhood of 2 of ray , the function shall result in an absolute value bigger than M: 0 M NOW IT S YOUR TURN N 2x 4 , Given the function y = _____ x2 4 verify that only the straight-line x = 2 is a vertical asymptote for the function. 126 2 O 2 2 2 x |x 3 2| > M _____ Since the absolute value is always positive for x 2, following the property of inequalities for real numbers, we can equally write: 3 3 3 |x 2| < __ 2 __ < x < 2 + __ M M M 3 This represents the neighborhood of 2 of ray = __ which verifies the limit. M The straight-line x = 2 is the vertical asymptote for the function.

Il Maraschini-Palma - volume 5
Il Maraschini-Palma - volume 5