Question

find the vertical, horizontal, or oblique asymptote for the equation. r(x)=x^2+x-6/x-3

Answer 1

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Answer 2

The vertical asymptotes occur for values of x that make the rational function undefined (i.e. the values of x that make the denominator = 0).
In this case, values of x for which the following holds: \(x-3=0\)

That is, assuming your function is
\[f(x)=\frac{x^2+x-6}{x-3}\]

Horizontal asymptotes occur at the values of y that are equal to the limits at infinity; in other words, a horizontal asymptote occurs at the lines \(y=L_1\text{ and }y=L_2\) if
\[\lim_{x\to\infty}f(x)=L_1,\text{ and }\lim_{x\to-\infty}f(x)=L_2\]

If these limits are not finite, the function has oblique/slant asymptotes that take on the form of the remainder of the quotient. That is, using long division, the oblique asymptote will be
\[y=\frac{R(x)}{x-3},\text{ where }R(x)\text{ is the remainder}.\]