# How do you find the vertical, horizontal or slant asymptotes for #(x² - 3x - 7)/(x+3) #?

The vertical asymptote is

and the slant asymptote is

graph{(y-((x^2-3x-7)/(x+3)))(y-x+6)=0 [-58.5, 58.5, -29.27, 29.28]}

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To find the vertical asymptote, set the denominator equal to zero and solve for (x). In this case, set (x + 3 = 0) and solve for (x). The result is (x = -3).

To find horizontal or slant asymptotes, you compare the degrees of the numerator and denominator polynomials:

- If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at (y = 0).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the two polynomials.
- If the degree of the numerator is greater than the degree of the denominator, there is a slant (oblique) asymptote.

In this case, the degree of the numerator is 2 and the degree of the denominator is 1, so there is no horizontal asymptote, but there could be a slant asymptote.

To find the slant asymptote, perform polynomial long division or use synthetic division to divide the numerator by the denominator. The quotient will be the equation of the slant asymptote.

After performing polynomial long division or synthetic division, you'll find the quotient to be (x - 6), which represents the equation of the slant asymptote.

So, the vertical asymptote is (x = -3) and the slant asymptote is (y = x - 6). There is no horizontal 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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