Vertical Asymptotes Vertical Asymptotes An Asymptote is basically a line in which a graph approaches, but it never intersects the line. For example: In the following given graph of P =1 / Q, (here the x axis is denoted by ‘P’ and y axis is denoted by Q) the line approaches the p-axis (Q=0), but the line never touches it. The line can reach to infinity but it never touch. The line will not actually reach Q = 0, but will always get closer and closer. Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.) In coordinate Geometry, an Asymptote of a curve is basically a line which approaches the curve arbitrarily so close that it tends to meet the curve at infinite, but it never intersects the line.

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In classical mathematics it was said that an asymptote and the curve never meets but modernly such as in algebraic geometry, an asymptote of a curve is a line which is Tangent to that curve at infinity. It shows 1/P, which has a Vertical Asymptote at P = 0 and a Horizontal Asymptote at Q = 0. The line approaches the P-axis (Q=0), infinitely close but it never touches the axis. A graph to show asymptote, here the x axis is denoted by ‘P’ and y axis is denoted by Q There are two types of asymptotes: 1. vertical asymptotes 2. Horizontal asymptotes. Let’s start with some Asymptote rules: As for vertical asymptotes they occur where the denominator equals zero. And for horizontal asymptotes, following rules are applicable: Horizontal asymptotes = Leading Coefficient / Leading Coefficient 1. If the degree of the numerator is greater than the degree of the denominator then there will be no horizontal asymptote. Even if it is greater by exactly 1 there can be a slant asymptote. Horizontal asymptotes = Denominator

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Example: f(x) =

11x3 /(x2 + 7)

limx→¥f(x) = ¥ limx→-¥f(x) = -¥ 2. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be the fraction of the leading coefficients. Example: f(x) = 5x / 3x + 5 3. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote must be the line y = 0. Example: f(x) = (x3 + 7x) /( x5 +18x4 –x)

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