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asymptotes of graph of rational function

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Let $f(x) = \frac{P(x)}{Q(x)}$ be a fractional expression where and are polynomials with real coefficients such that their quotient can not be reduced to a polynomial. We suppose that and have no common zeros.

If the division of the polynomials is performed, then a result of the form

$\displaystyle f(x) = H(x)+\frac{R(x)}{Q(x)}$

is gotten, where

and

are polynomials such that

$\displaystyle \deg{R(x)} < \deg{Q(x)}$

Every zero of the denominator gives a vertical asymptote .
If $\deg{H(x)} < 1$ (i.e. 0 or $-\infty$ ) then the graph has the horizontal asymptote .
If $\deg{H(x)} = 1$ then the graph has the skew asymptote .

Proof of 2 and 3. We have $f(x)-H(x) = \frac{R(x)}{Q(x)}\,\to 0$ as $\vert x\vert\to\infty$ .

Remark. Here we use the convention that the degree of the zero polynomial is $-\infty$ .

"asymptotes of graph of rational function" is owned by eshyvari. [ owner history (1) ]

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AMS MSC:	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	26C15 (Real functions :: Polynomials, rational functions :: Rational functions)
	51N99 (Geometry :: Analytic and descriptive geometry :: Miscellaneous)

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