# How do you find the asymptotes for #y = x/(x-6)#?

vertical asymptote x = 6

horizontal asymptote y = 1

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

solve: x - 6 = 0 → x = 6 is the equation.

If the degree of the numerator and denominator are equal , as in this question , both of degree 1 , then the equation can be found by taking the ratio of leading coefficients.

here is the graph of the function as an illustration. graph{x/(x-6) [-20, 20, -10, 10]}

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To find the asymptotes of ( y = \frac{x}{x-6} ):

- Vertical asymptote: Set the denominator equal to zero and solve for ( x ). The vertical asymptote(s) occur where the denominator is zero.

[ x - 6 = 0 ]

[ x = 6 ]

So, there is a vertical asymptote at ( x = 6 ).

- Horizontal asymptote: Compare the degrees of the numerator and the denominator.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ).

If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In this case, since the degrees of both the numerator and the denominator are 1, the horizontal asymptote is the ratio of the leading coefficients:

[ y = \frac{1}{1} ]

So, the horizontal asymptote is ( y = 1 ).

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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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