How do you find the vertical, horizontal or slant asymptotes for #-7 / (x+4)#?

Answer 1

See below.

Vertical asymptotes occur where the function is undefined.

#-7/(x+4)#

For #x=-4# #-7/(x+4)=-7/(0)# undefined.

The line #color(blue)(x=-4)# is a vertical asymptote:

as # x->oo# , #color(white)(888)-7/(x+4)->0#

as # x->-oo# , #color(white)(888)-7/(x+4)->0#

The x axis is a horizontal asymptote:

There is no oblique asymptote. Oblique asymptotes occur when the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator.

Graph:

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Answer 2

To find the vertical, horizontal, or slant asymptotes for the function ( f(x) = \frac{-7}{x + 4} ), follow these steps:

  1. Vertical Asymptote: Vertical asymptotes occur where the function is undefined, which happens when the denominator equals zero. Set ( x + 4 = 0 ) and solve for ( x ).

[ x + 4 = 0 ] [ x = -4 ]

So, the vertical asymptote is at ( x = -4 ).

  1. Horizontal Asymptote: Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the denominator is greater, there is no horizontal asymptote.

  2. Slant Asymptote: If the degree of the numerator is exactly one more than the degree of the denominator, a slant asymptote may exist. To find it, perform polynomial long division or use synthetic division to divide the numerator by the denominator.

In this case, when we divide (-7) by (x + 4), we get a quotient of (-7) and a remainder of 0. Therefore, the slant asymptote is the quotient, which is ( y = -7 ).

In summary:

  • Vertical asymptote: ( x = -4 )
  • Horizontal asymptote: None
  • Slant asymptote: ( y = -7 )
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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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