Horizontal, and Oblique Asymptotes - Maple Help

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Horizontal, and Oblique Asymptotes

Main Concept

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.

Vertical Asymptotes

 Vertical Asymptote A vertical asymptote is a vertical line, $x=a$, that has the property that either:     or     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  has a vertical asymptote at $x=0$.

Horizontal Asymptotes

 Horizontal Asymptote A horizontal asymptote is a horizontal line, $y=a$, that has the property that either:       or     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 ${x}^{n}$-th term of the numerator (it may be zero if the numerator has a smaller degree) by the coefficient of the ${x}^{n}$-th term of the denominator. For example, the function  goes to 7 as $x$ approaches $±\infty$.

Oblique Asymptotes

 Oblique Asymptote An oblique or slant asymptote is an asymptote along a line $y=\mathrm{mx}+b$, where . 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  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  and see how they affect the horizontal and oblique asymptotes.

 $n:$ $a:$ $k:$

$f\left(x\right)=$



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