# How do you find vertical, horizontal and oblique asymptotes for #(x^2-5x+6)/(x-4)#?

The vertical asymptote is

The oblique asymptote is

No horizontal asymptote

Let's do a long division

Therefore,

graph{(y-(x^2-5x+6)/(x-4))(y-x+1)(y-100(x-4))=0 [-18.3, 17.74, -6.74, 11.28]}

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To find the vertical asymptote(s) of a rational function, set the denominator equal to zero and solve for x. In this case, the vertical asymptote occurs when x - 4 = 0, yielding x = 4.

To find the horizontal asymptote, 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 degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of both polynomials. In this case, since both the numerator and denominator have the same degree (1), the horizontal asymptote is y = coefficient of x in numerator / coefficient of x in denominator, which is 1/1 = 1.

To find oblique asymptotes, divide the numerator by the denominator using long division or synthetic division. The quotient obtained will represent the equation of the oblique asymptote, if it exists.

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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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