SOLUTION: Which one of the following is true? (1) The function, f(x)=2x^2+1/x^2-1, does not have a horizontal asymptote. (2) The graph of a rational function cannot have x-intercepts. (3) Th
Algebra.Com
Question 202900: Which one of the following is true? (1) The function, f(x)=2x^2+1/x^2-1, does not have a horizontal asymptote. (2) The graph of a rational function cannot have x-intercepts. (3) The horizontal asymptote for the graph of y=4x-1/x+3 is x=-3. (4) The graph of f(x)=2x+4/x^2-4 has only one vertical asymptote. THIS IS A TOUGH ONE....HELP.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
1) Has a horizontal asymptote at 
because the numerator and denominator are of the same degree, hence there is a horizontal asymptote at the value of the lead coefficient of the numerator divided by the lead coefficient of the denominator. This one is false.
2) The graph of a rational function can have 
intercepts. The graph will intercept the -axis at any zero of the numerator polynomial that is not also a zero of the denominator polynomial. Consider 
. This one is false.
3) As in #1, the horizontal asymptote is at 
where 
is the value of the lead coefficient of the numerator divided by the lead coefficient of the denominator. Hence, the horizontal asymptote is at 
not 
. This one is false.
4) This one looks false on the face of it, because you would expect a vertical asymptote at any zero of the denominator polynomial. However, note that -2 is a zero of the denominator, but it is also a zero of the numerator. Hence, while there is a discontinuity at -2, there is no asymptote. In order to prove it, you need a little trick from the Calculus called L'Hôpital's Rule. It says that if:
 = 0\text{ or }\pm\infty)
and
= 0\text{ or }\pm\infty)
and if
}{g'(x)})
exists, then
}{g(x)} = \lim_{x\rightarrow c}\ \frac{f'(x)}{g'(x)})
So for the function given in #4,
 = 2)
and
 = 2x)
and
}{g'(x)} = \frac{2}{2(-2)}=-\frac{1}{2})
not 
as would be the case if there were an asymptote at that point.
Therefore #4 is the true statement.
John
RELATED QUESTIONS
Find the equations for the horizontal and vertical asymptotes of the following. Type... (answered by stanbon)
Find the equations for the horizontal and vertical asymptotes of the following. Type none (answered by stanbon)
Find the equations of the horizontal and vertical asymptotes for the following. Tye none (answered by jim_thompson5910)
The graph of the rational function:
f(x)= x+1/x^2-3x+2
has a horizontal asymptote with... (answered by stanbon)
Which of the following is true of f(x)?
f(x)= (x^2+8x+15)/(x^2-2x-15).
One of the... (answered by MathLover1)
Find the horizontal and vertical asymptotes of the following. Type none if the function... (answered by stanbon)
Graph the Rational Function & give the equations of its asymptotes.
(answered by jsmallt9)
Find the horizontal and vertical asymptotes of the following. Type “none” if the function (answered by jim_thompson5910)
Find the equations of the horizontal and vertical asymptotes for the following. Type none (answered by solver91311)