# How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)= x/(x(x-2))#?

vertical asymptote x = 2

horizontal asymptote y = 0

The first step here is to simplify f(x) by cancelling the x.

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

solve : x - 2 = 0 → x = 2 is the asymptote

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 hence there are no oblique asymptotes. graph{1/(x-2) [-10, 10, -5, 5]}

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To find the vertical asymptotes of the function ( f(x) = \frac{x}{x(x-2)} ), set the denominator equal to zero and solve for ( x ). In this case, the denominator ( x(x-2) ) equals zero when ( x = 0 ) or ( x = 2 ). Therefore, the vertical asymptotes are ( x = 0 ) and ( x = 2 ).

To find the horizontal asymptote, examine the behavior of the function as ( x ) approaches positive or negative infinity. In this case, since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ).

To find the oblique asymptote, perform polynomial division or long division to divide the numerator by the denominator. The quotient of this division represents the oblique asymptote, if it exists. For ( f(x) = \frac{x}{x(x-2)} ), performing the division yields ( 1 + \frac{2}{x-2} ). As ( x ) approaches infinity or negative infinity, the term ( \frac{2}{x-2} ) approaches zero, leaving ( y = 1 ) as the 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.

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