An asymptote is a line that the graph of a function approaches as either x or y approaches infinity. There are three types of asymptotes: vertical, horizontal and oblique.
A vertical asymptote is a vertical line, x=a, that has the property that either:
1. limx→a−fx= ±∞
2. limx→a+fx = ±∞
That is, as x approaches a from either the positive or negative side, the function approaches infinity.
Vertical asymptotes occur at the values where a rational function has a denominator of 0. The function is undefined at these points.
A horizontal asymptote is a horizontal line, y=a, that has the property that either:
2. limx→−∞fx =a
Horizontal asymptotes occur when the numerator of a rational function has degree less than or equal to the degree of the denominator. If the denominator has degree n, the horizontal asymptote can be calculated by dividing the coefficient of the xn-th term of the numerator (it may be 0 if the numerator has a smaller degree) by the coefficient of the xn-th term of the denominator.
An oblique or slant asymptote is an asymptote along a line y=mx+b, where 0<m<∞. Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.
Click or drag to place up to 5 points through which a curve (blue) will be drawn. The x-intercepts (green), the reciprocal of the curve (black) and any vertical asymptotes of the reciprocal (magenta) will also be shown.
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