How do you find the vertical, horizontal or slant asymptotes for #f(x) = (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 let the denominator equal zero.
divide terms on numerator/denominator by x
This is the graph of f(x). graph{(2x+3)/(3x+1) [10, 10, 5, 5]}
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To find the vertical, horizontal, or slant asymptotes for ( f(x) = \frac{2x + 3}{3x + 1} ), follow these steps:

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function equals zero but the numerator does not. Set the denominator equal to zero and solve for ( x ). The values of ( x ) obtained are the vertical asymptotes.

Horizontal Asymptotes: To find horizontal asymptotes:
 If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line ( y = 0 ).
 If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
 If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Slant Asymptotes (Oblique Asymptotes): Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. To find the equation of the slant asymptote:
 Perform polynomial long division or synthetic division to divide the numerator by the denominator.
 The quotient obtained represents the equation of the slant asymptote.
Following these steps, you can determine the vertical, horizontal, or slant asymptotes of the given function ( f(x) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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