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

Question

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

Answer 1

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.

Answer 2

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...