How do you find the asymptotes for #y=(x-3)/(2x+5)#?

Answer 1

Vertical Asymptote: #x=-5/2#
Horizontal Asymptote: #y=1/2#

For the vertical asymptote:

Equate the denominator to zero, then solve for x #2x+5=0# #2x=-5# #x=-5/2#

For the horizontal asymptote:

Take the limit of the function

#lim_(xrarr oo) y=lim_(xrarr oo) (x-3)/(2x+5)=1/2#

and therefore

#y=1/2# is a horizontal asymptote

graph{(y- (x-3)/(2x+5))(y-1/2)=0[-20,20,-10,10]}

God bless....I hope the explanation is useful.

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Answer 2

To find the asymptotes of the rational function y = (x - 3) / (2x + 5), you need to determine where the function approaches infinity or negative infinity.

  1. Horizontal Asymptote:

    • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 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.
  2. Vertical Asymptote:

    • Set the denominator equal to zero and solve for x. The vertical asymptote(s) occur where the function is undefined due to division by zero.

For the given function y = (x - 3) / (2x + 5):

  1. Degree of the numerator: 1
  2. Degree of the denominator: 1
  3. Leading coefficients: 1 for both numerator and denominator

Since the degrees are equal and the leading coefficients are also equal, divide the leading coefficients: Horizontal Asymptote: y = 1/2

For vertical asymptotes, set the denominator equal to zero and solve for x: 2x + 5 = 0 x = -5/2

So, the vertical asymptote is x = -5/2.

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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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