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Determine the asymptotes for p(x)=-4x-1/x-5

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2 Answers

Find the asymptotes for p(x) = (-4x-1)/(x-5)
 
The denominator, (x-5), equals zero at x=5.  So there will be a vertical asymptote at x=5.
 
When the degree of the numerator and denominator are the same, there will be a horizontal asymptote that is equal to the ratio of the coefficients of the highest degree terms.  The "degree" refers to the highest exponent.  In this case, it's x1, or just x, in both the numerator and the denominator.  The coefficient of the x term in the numerator is -4, and in the denominator it's 1.  The horizontal asymptote, then, is y = (-4)/1 = -4.
Perhaps this function should be written: p(x)=(-4x-1)/(x-5)
If this is the case, then, what is the vertical asymptote (VA)?  
Remember... the denominator can not be equal to 0.  So...
what value of x will make x-5=0? 
That vertical line will be your VA: x=?
Now, the horizontal asymptote (HA)...
Please review the 3 cases for defining HA:
1) When it does exist. 2) When it doesn't exist; and 3) When HA is the x-axys or y=0
In this function... Does it exist?  
Yes it does and it will be defined by the leading coefficients -4 and 1:
y=-4/1.  So, the HA is y=-4
Now, please write your answers and let me know...