# How do you find vertical, horizontal and oblique asymptotes for #f(x) = (3x + 5) /( x - 2)#?

vertical asymptote at x = 2

horizontal asymptote at y = 3

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

Horizontal asymptotes occur as

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{(3x+5)/(x-2) [-20, 20, -10, 10]}

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To find the vertical asymptote of the function ( f(x) = \frac{3x + 5}{x - 2} ), set the denominator equal to zero and solve for ( x ). The vertical asymptote occurs where the function is undefined, which happens when the denominator is zero:

[ x - 2 = 0 ] [ x = 2 ]

So, the vertical asymptote is ( x = 2 ).

To find horizontal asymptotes, determine the limit of the function as ( x ) approaches positive or negative infinity. If the limit exists, it represents the horizontal asymptote.

[ \lim_{x \to \infty} \frac{3x + 5}{x - 2} = 3 ] [ \lim_{x \to -\infty} \frac{3x + 5}{x - 2} = 3 ]

Since the limit exists and is finite, the horizontal asymptote is ( y = 3 ).

For oblique asymptotes, divide the numerator by the denominator using polynomial long division or synthetic division. In this case, dividing ( 3x + 5 ) by ( x - 2 ) gives:

[ 3 ]

So, the oblique asymptote is ( y = 3 ).

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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.

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