9514 1404 393
Answer:
True (or False--see comment)
Step-by-step explanation:
No restrictions were placed on the nature of the rational function, so it is possible to create one that has all three kinds of asymptotes.
The attached graph shows that, along with the rational function whose graph it is. The function's asymptotes are ...
- horizontal asymptote: y = 0
- vertical asymptote: x = -4
- slant asymptote: y = x -4
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Additional comment
Our example function makes use of an exponential function. While that can be represented as a polynomial of infinite degree, it would usually not be considered a polynomial. Most authors define a rational function to be the ratio of polynomials.
A ratio of polynomials of finite degree cannot have all three kinds of asymptotes. It can have a vertical asymptote with either a horizontal or slant asymptote, but cannot have both a horizontal and a slant asymptote. The asymptote function is the quotient from the division of the numerator polynomial by the denominator polynomial. Any remainder from that division will necessarily tend to zero as the independent variable gets large.
