Question 1204539
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Answers:
(a) <font color=red>True</font>
(b) <font color=red>False</font>
(c) <font color=red>False</font>
(d) <font color=red>True</font>
(e) <font color=red>True</font>


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Explanation for part (a)


Let,
n = degree of the numerator
d = degree of the denominator
where degree = largest exponent of a single variable polynomial


If n < d, then this will mean the denominator polynomial races faster to infinity compared to the numerator polynomial. 


As such, the denominator gets much larger and the overall fraction approaches 0. 
An example would be {{{y = (x+5)/(x^2-7x)}}}


Use a graphing tool such as a TI83, Desmos, or GeoGebra to plot out that equation. You should notice the curve steadily approaches the x axis. It will never actually arrive there.


Or you can use technology to create a table of values to see what happens to the value of y when x gets really large.
For very large values of x, the y values should be really small or 0 depending on how things are rounded. As mentioned, the y value never actually will equal 0 itself.
 

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Explanation for part (b)


Again we define the following
n = degree of the numerator
d = degree of the denominator


If n = d+1, then we'll have an oblique asymptote or slant asymptote. 


An example would be {{{y = (5x^3-2)/(x^2+4)}}}, in which case the slant asymptote is {{{y = 5x}}}


For other cases when n > d+1, the curve's endpoints will approach some other curve. 
Meaning the asymptote is no longer a straight line.
Some textbooks would consider the asymptote to be "none", but I think a curve can be an asymptote.


For example, {{{y = (x^4-7x^2)/(x+5)}}} has its the endpoints approach the cubic curve {{{y = x^3-5x^2+18x-90}}}. 
This can be determined using polynomial long division or synthetic division. 
Synthetic division can be used only when the degree of the denominator is 1.


WolframAlpha can be used to determine the asymptotes.
For example
<a href="https://www.wolframalpha.com/input?i=asymptotes+%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29">https://www.wolframalpha.com/input?i=asymptotes+%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29</a>
The search input was "<font color=red>asymptotes (2x^3 + 4x^2 - 9)/(3 - x)</font>" without quotes.
The "vertical asymptote" section is basically saying the vertical asymptote is x = 3.
The "parabolic asymptote" section states that {{{(2x^3 + 4x^2 - 9)/(3 - x)}}} is asymptotic to  {{{-2x^2 - 10x - 30}}}
WolframAlpha also provides a graph along with the asymptotes as dashed lines/curves.


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Explanation for part (c)


n = degree of the numerator
d = degree of the denominator


If n = d, then the horizontal asymptote is {{{y = a/b}}} where 'a' is the leading coefficient of the numerator, while b is the leading coefficient of the denominator.


For example, the horizontal asymptote of {{{y = (3x+26)/(7x+4)}}} is {{{y = 3/7}}}


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Explanation for part (d)


This statement is true. 
It can be rephrased to "as x goes to positive or negative infinity, the value of y gets closer to 'a'"


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Explanation for part (e)


The wording "guiding line" is a bit strange, but I'm assuming it refers to the idea the curve steadily approaches this line as x gets really big or really small.


For some rational functions, the curve will cross over the horizontal asymptote. Perhaps more than once. Other rational functions will not have the curve ever touch the horizontal asymptote at all.


For either case, as x goes off to infinity, the curve gets closer to this line. 


Further Reading
<a href="https://www.mathsisfun.com/algebra/asymptote.html">https://www.mathsisfun.com/algebra/asymptote.html</a>
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