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Let’s develop a simple method to find infinite limits and horizontal asymptotes. Here are 3 functions that appear to look almost the same, but there are subtle differences. 6x 2 y 2 2x 4 6x 2 y 2 2x 4 2 6x 2 y 2 2x 4 3 Let’s explore each as x approaches ∞ lim x 6x 2 2 2x 4 Look at the degree of each polynomial The degree of the bottom, 2, is greater than the degree of the top, 1. As x grows without bound, the bottom will dominate and the limit will go to 0 lim x 6x 2 0 2 2x 4 6x 2 y 2 2x 4 2 Here, the degree of the top is equal to the degree of the bottom (both are 2) The limit will be the ratio of the leading coefficients (the coefficients of the terms of highest degree). 6x 2 6 3 2 2x 4 2 2 lim x 6x 2 y 2 2x 4 3 Here, the degree of the top, 3, is greater than the degree of the bottom, 2. The numerator will dominate and this limit will grow without bound to infinity. 6x 2 2 2x 4 3 lim x We can quickly find the horizontal asymptotes: 6x 2 y 2 2x 4 y = 0, same as the limit, this is the x-axis 6x 2 y 2 2x 4 y = 3, a horizontal line 6x 2 y 2 2x 4 No horizontal asymptote, the function grows without bound and does not approach a single value 2 3 Here is a quick quiz for you. Find the horizontal asymptotes: 5 x 3 7 x 3 The degrees are the y 3 3x 2 x same (3) so y = 5/3 4x 6x y 2 3x 5 x 2 4 The degree of the top is greater (4 > 2) so there is no horizontal asymptote