Horizontal, and Oblique Asymptotes
An asymptote is a line that the graph of a function approaches as either x or y go to positive or negative 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:
limx→a+fx = ±∞
That is, as x approaches a from either the positive or negative side, the function approaches positive or negative infinity.
Vertical asymptotes occur at the values where a rational function has a denominator of zero. The function is undefined at these points because division by zero mathematically ill-defined. For example, the function fx = 1x has a vertical asymptote at x=0.
A horizontal asymptote is a horizontal line, y=a, that has the property that either:
This means, that as x approaches positive or negative infinity, the function tends to a constant value 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 zero if the numerator has a smaller degree) by the coefficient of the xn-th term of the denominator. For example, the function fx = x2+1x3+7 goes to 7 as x approaches ±∞.
An oblique or slant asymptote is an asymptote along a line y=mx+b, where m≠0. Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.
For example, the function fx = x+1x has an oblique asymptote about the line y=x and a vertical asymptote at the line x=0.
Use the sliders to choose the values a, n, and k in the equation fx=a⋅xn+k2 x2−x−1 and see how they affect the horizontal and oblique asymptotes.
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