How do you find the vertical, horizontal or slant asymptotes for # f(x) = (3x - 12 ) / ( x + 4) #?

Answer 1

vertical asymptote x = -4
horizontal asymptote y = 3

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : x + 4 = 0 → x = -4 is the asymptote

Horizontal asymptotes occur as

#lim_(xto+-oo),f(x)toc" (a constant)"#

divide terms on numerator/denominator by x

#((3x)/x-12/x)/(x/x+4/x)=(3-12/x)/(1+4/x)#
as #xto+-oo,f(x)to(3-0)/(1+0)#
#rArry = 3/1=3" is the asymptote"#

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1). Hence there are no slant asymptotes. graph{(3x-12)/(x+4) [-20, 20, -10, 10]}

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Answer 2

To find the vertical asymptote of ( f(x) = \frac{3x - 12}{x + 4} ):

Set the denominator equal to zero and solve for ( x ):

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

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

To find the horizontal asymptote:

Compare the degrees of the numerator and the denominator:

The degree of the numerator is 1, and the degree of the denominator is also 1.

The horizontal asymptote is given by the ratio of the leading coefficients:

[ y = \frac{3}{1} ] [ y = 3 ]

So, ( y = 3 ) is the horizontal asymptote.

To check for a slant (oblique) asymptote:

If the degree of the numerator is one more than the degree of the denominator, then there is a slant asymptote.

Here, the degree of the numerator is 1, and the degree of the denominator is also 1. Therefore, there is no slant asymptote for this function.

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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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