The function f(x)=(4x+1)/x+3. Where is the hole and horizontal asymptote

Question

Func, stats and trig 

Answer 1

Holes occur at zeros of cancelled factors. there  no factors to cancel so there are NO HOLES

 

there will however be a vertical asymptote at the zero of the denominator. At x=-3

 

vertical asymptotes and holes are not the same.

although they are both discontinuities they are different.

 

a vertical asymptote is an example of a no removable discontinuity. A hole is a removable discontinuity.

 

horizontal asymptote so exist when the degree of the numerator = degree of denominator

 

when This occurs the horizontal asymptote= ratio of lead coefficients = 4/1= 4. The horizontal asymptote is y=4

 

you also can get a horizontal asymptote when the degree of the denominator is larger than the numerator. In those instances the horizontal asymptote is always y=0

Answer 2

f(x) = (4x+1) / (x+3)

 

1. Where is the hole: The hole is the quantity for which f(x) is undefined.  This occurs if the denominator equals zero, i.e. x + 3 = 0.  Solving for x gives x = -3

 

2. What is horizontal asymptote": The horizontal asymptote is the number that f(x) approaches when x becomes an increasingly large (positive or negative) number.

  When |x| is very large, then the numerator will be dominated by the 4x term.  

  When |x| is very large, the denominator will be dominated by the x term.

 

Then f(x) can be approximated by: (4x) / (x) = 4

 

So, f(x) approaches 4 as x -> +/- infinity, and therefore the horizontal asymptote is f(x) = 4