How do you find the vertical, horizontal or slant asymptotes for #(x^2-4)/(x^3+4x^2)#?

Answer 1

The vertical asymptotes are #x=0# and #x=-4#
No slant asymptotes.
The horizontal asymptote is #y=0#

Let #f(x)=(x^2-4)/(x^3+4x^2)#
Let's factorise the denominator #x^3+4x^2=x^2(x+4)#
The domain of #f(x)# is #D_f(x)=RR-{0,-4}#
As we cannot divide by #0#
So #x!=0# and #x!=-4#
The vertical asymptotes are #x=0# and #x=-4#
As the degree of the numerator is #<# the degree of the denominator, there is no slant asymptotes.

For calculating the limits, we take the terms of highest degree in the numerator and the denominator

#lim_(x->-oo)f(x)=lim_(x->-oo)x^2/x^3=lim_(x->-oo)1/x=0^(-)#
#lim_(x->+oo)f(x)=lim_(x->+oo)x^2/x^3=lim_(x->+oo)1/x=0^(+)#
So, #y=0# is a horizontal asymptote

graph{(x^2-4)/(x^3+4x^2) [-10, 10, -5, 5]}

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Answer 2

To find the vertical, horizontal, or slant asymptotes for the function ( \frac{x^2 - 4}{x^3 + 4x^2} ), follow these steps:

  1. Vertical Asymptotes: Set the denominator equal to zero and solve for ( x ). Vertical asymptotes occur where the function approaches positive or negative infinity as ( x ) approaches the value(s) found.

  2. Horizontal Asymptotes: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degrees are equal, divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

  3. Slant Asymptotes (Oblique Asymptotes): If the degree of the numerator is exactly one more than the degree of the denominator, perform long division to find the quotient and remainder. The quotient represents the equation of the slant asymptote.

Apply these steps to the given function to determine its vertical, horizontal, and slant asymptotes.

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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.

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