document.write( "Question 1135352: I know how to find vertical asymptotes of rational functions but is there a set guide of rules for Horizontal asymptotes?\r
\n" ); document.write( "\n" ); document.write( "the internet has a lot of different rules.\r
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\n" ); document.write( "\n" ); document.write( "please and thank you \n" ); document.write( "

Algebra.Com's Answer #752974 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "(1) If the degree of the numerator is less than the degree of the denominator, then for very large positive or very large negative x the denominator gets big faster than the numerator, so the value of the rational function goes towards 0; the horizontal asymptote is y=0.

\n" ); document.write( "Example: (x+1)/(x^2+2x+2); horizontal asymptote y=0

\n" ); document.write( " $\"graph%28400%2C200%2C-10%2C10%2C-1%2C1%2C%28x%2B1%29%2F%28x%5E2%2B2x%2B2%29%29\"$

\n" ); document.write( "(2) If the degree of the numerator is the same as the degree of the denominator, then for very large positive or very large negative x the numerator and denominator get larger at the same rate. And for very large positive or negative x, the lower degree terms are insignificant compared to the leading term. So in this case the horizontal asymptote is the ratio of the leading terms -- which is the same as saying it is the ratio of the coefficients of the leading terms.

\n" ); document.write( "Example: (3x+2)/(x-1); horizontal asymptote y=3

\n" ); document.write( " $\"graph%28400%2C200%2C-10%2C10%2C-5%2C5%2C%283x%2B2%29%2F%28x-1%29%2C3%29\"$
\n" ); document.write( "(3) If the degree of the numerator is greater than the degree of the denominator, then for very large positive or very large negative x the numerator get big faster than the denominator, so the value of the rational function grows larger positive or larger negative; there is no horizontal asymptote.

\n" ); document.write( "Example: (x^2+3x+1)/(x+2); no horizontal asymptote (slant asymptote y=x+1)

\n" ); document.write( " $\"graph%28400%2C200%2C-5%2C5%2C-10%2C10%2C%28x%5E2%2B3x%2B1%29%2F%28x%2B2%29%2Cx%2B1%29%29\"$

\n" ); document.write( "In this example, the rational function is equivalent to $\"%28x%2B1%29-1%2F%28x%2B2%29\"$ . As x gets very large positive or very large negative, the remainder becomes insignificant, and the function value gets very close to the value of x+1. So the graph of y=x+1 is a SLANT asymptote of this rational function.

\n" ); document.write( "QUICK SUMMARY:

\n" ); document.write( "degree of numerator smaller --> horizontal asymptote y=0
\n" ); document.write( "degree of numerator = degree of denominator --> horizontal asymptote is the ratio of the leading coefficients
\n" ); document.write( "degree of numerator larger --> no horizontal asymptote \n" ); document.write( "

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