How do you identify all asymptotes or holes for #f(x)=x/(-4x^2-16x)#?

Answer 1

#"vertical asymptote at "x=-4#
#"horizontal asymptote at "y=0#

#"simplify f(x) by factorising the denominator"#
#f(x)=cancel(x)/(-4cancel(x)(x+4))=-1/(4(x+4))#
#"the removal of the factor x from the numerator/denominator"# #"indicates a hole at x = 0"#

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

#"solve "4(x+4)=0rArrx=-4" is the asymptote"#
#"horizontal asymptotes occur as "#
#lim_(xto+-oo),f(x)toc" ( a constant)"#
#"divide terms on numerator/denominator by x"#
#f(x)=-(1/x)/((4x)/x+16/x)=-(1/x)/(4+16/x)#
as #xto+-oo,f(x)to0/(4+0)#
#rArry=0" is the asymptote"# graph{x/(-4x^2-16x) [-10, 10, -5, 5]}
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To identify all asymptotes or holes for the function ( f(x) = \frac{x}{-4x^2 - 16x} ), we first need to analyze the behavior of the function as ( x ) approaches certain values.

  1. Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. To find them, we solve the equation ( -4x^2 - 16x = 0 ) for ( x ). This yields the values ( x = 0 ) and ( x = -4 ). Therefore, the vertical asymptotes are at ( x = 0 ) and ( x = -4 ).

  2. Horizontal Asymptotes: Horizontal asymptotes occur when ( x ) approaches positive or negative infinity. To find them, we examine the behavior of the function as ( x ) approaches infinity. By observing the highest power terms in the numerator and denominator, we can see that as ( x ) approaches infinity, the term ( -4x^2 ) in the denominator dominates. Thus, the horizontal asymptote is at ( y = 0 ).

  3. Holes: Holes occur when there is a factor common to both the numerator and the denominator of the function, resulting in a removable discontinuity. To find them, we factor both the numerator and the denominator and cancel any common factors. In this case, the numerator ( x ) and the denominator ( -4x(x + 4) ) share a common factor of ( x ). Canceling this common factor, we get ( f(x) = \frac{1}{-4(x + 4)} ). This function has no real zeros in the numerator that would cause a hole. Hence, there are no holes in this function.

Therefore, the identified asymptotes for the function ( f(x) = \frac{x}{-4x^2 - 16x} ) are:

  • Vertical asymptotes at ( x = 0 ) and ( x = -4 ).
  • Horizontal asymptote at ( y = 0 ).
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7