How do you find vertical, horizontal and oblique asymptotes for #(2x+3)/(3x+1) #?

Answer 1

vertical asymptote #x=-1/3#
horizontal asymptote #y=2/3#

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

solve : 3x + 1 = 0 #rArrx=-1/3" is the asymptote"#
Horizontal asymptotes occur as #lim_(xto+-oo) , y to 0#

divide terms on numerator/denominator by x

#((2x)/x+3/x)/((3x)/x+1/x)=(2+3/x)/(3+1/x)#
as #xto+-oo ,yto(2+0)/(3+0)#
#rArry=2/3" is the asymptote"#

Oblique 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 oblique asymptotes. graph{(2x+3)/(3x+1) [-10, 10, -5, 5]}

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator. 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 of both polynomials to find the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote. For oblique asymptotes, divide the numerator by the denominator using long division or polynomial division. The quotient represents the oblique asymptote.

For ( \frac{2x+3}{3x+1} ), there is no vertical asymptote. The degrees of the numerator and denominator are both 1, so the horizontal asymptote is ( y = \frac{2}{3} ). There is no oblique asymptote.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7