SOLUTION: I know how to find vertical asymptotes of rational functions but is there a set guide of rules for Horizontal asymptotes? the internet has a lot of different rules. please

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Question 1135352: I know how to find vertical asymptotes of rational functions but is there a set guide of rules for Horizontal asymptotes?
the internet has a lot of different rules.

please and thank you
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
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I know how to find vertical asymptotes of rational functions but is there a set guide of rules for Horizontal asymptotes?
the internet has a lot of different rules.
...
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You do not need "internet" to get that help. Try a college algebra instructional book.

A vertical line which a function approaches but never passes through, is a vertical asymptote.

EXAMPLE
f%28x%29=6%2F%28x%2B9%29

x can NEVER be -9 because this function f is undefined for x at -9; the denominator would be 0, and this is not permitted.

On the other hand, x can be any value no matter how close-to or far from -9, and as x gets more to -9, f becomes increasingly toward either negative or to positive infinity. What stands in between the two parts of the function is this line x=-9. f never touches nor crosses it anywhere.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Writing by  @josgarithmetic is  IRRELEVANT  to the question in the post. 


The simple answer to the question of the post is THIS :


    If the given function has a finite value limit at x --> infinity  or at x --> -infinity,  

    then it defines the horizontal asymptote/asymptotes  y = limit.

Do you fully understand my answer ?


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(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.

Example: (x+1)/(x^2+2x+2); horizontal asymptote y=0

graph%28400%2C200%2C-10%2C10%2C-1%2C1%2C%28x%2B1%29%2F%28x%5E2%2B2x%2B2%29%29

(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.

Example: (3x+2)/(x-1); horizontal asymptote y=3

graph%28400%2C200%2C-10%2C10%2C-5%2C5%2C%283x%2B2%29%2F%28x-1%29%2C3%29
(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.

Example: (x^2+3x+1)/(x+2); no horizontal asymptote (slant asymptote y=x+1)

graph%28400%2C200%2C-5%2C5%2C-10%2C10%2C%28x%5E2%2B3x%2B1%29%2F%28x%2B2%29%2Cx%2B1%29%29

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.

QUICK SUMMARY:

degree of numerator smaller --> horizontal asymptote y=0
degree of numerator = degree of denominator --> horizontal asymptote is the ratio of the leading coefficients
degree of numerator larger --> no horizontal asymptote