How do you find vertical, horizontal and oblique asymptotes for #(2x+3)/(3x+1) #?
vertical asymptote
horizontal asymptote
Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.
divide terms on numerator/denominator by x
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]}
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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.
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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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