How to find the asymptotes of #f(x) = (x+1) / (x^2 +3x - 4)# ?

Answer 1

vertical asymptotes x = -4 , x = 1
horizontal asymptote y = 0

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

solve : #x^2+3x-4=0rArr(x+4)(x-1)=0#
#rArrx=-4" and " x=1" are the asymptotes"#

Horizontal asymptotes occur as

#lim_(xto+-oo),f(x)toc" (a constant)"#
divide terms on numerator/denominator by the highest power of the variable , in this case #x^2#
#(x/x^2+1/x^2)/(x^2/x^2+(3x)/x^2-4/x^2)=(1/x+1/x^2)/(1+3/x-4/x^2)#
as #xto+-oo,f(x)to(0+0)/(1+0-0)#
#rArry=0" is the asymptote"# graph{(x+1)/(x^2+3x-4) [-10, 10, -5, 5]}
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Answer 2

To find the asymptotes of the function ( f(x) = \frac{x + 1}{x^2 + 3x - 4} ), follow these steps:

  1. Determine the vertical asymptotes by identifying the values of ( x ) for which the denominator is equal to zero.
  2. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator of the rational function.

Vertical asymptotes occur at the values of ( x ) where the denominator ( x^2 + 3x - 4 ) equals zero. Solve ( x^2 + 3x - 4 = 0 ) to find these values.

Horizontal asymptotes can be determined by comparing the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degree of the numerator is equal to the degree of the denominator, 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.

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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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