SOLUTION: While graphing rational functions, when there are 2 vertical asymptotes and a horizontal symptote (giving 6 sectors), one of the three segments of the resulting graph seem to go th
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Question 1163632: While graphing rational functions, when there are 2 vertical asymptotes and a horizontal symptote (giving 6 sectors), one of the three segments of the resulting graph seem to go through the horizontal asymptote (some times once and at other times twice). And no segment of the graph seems to cross the vertical asymptote, however.
Question: So why do we have horizontal asymptotes that seem to be not an absolute barrier or are there any conditions in general when can you expect a segment of the graph cross the horizontal asymptote. Found 3 solutions by solver91311, Edwin McCravy, greenestamps:Answer by solver91311(24713) (Show Source):
Horizontal asymptotes describe the behavior of the function when the value of the independent variable either increases or decreases without bound. So the fact that the function is asymptotic to a given horizontal line does not mean that the value of the function cannot assume the function value of the horizontal line, it just means that the function will have that horizontal line function value as a limit whenever x gets really large in either direction.
So graph your function and find the smallest (most negative value) where the function intersects the line where is the lead coefficient on the numerator and is the lead coefficient on the denominator (for numerator and denominator of equal degree) or (for numerator of lesser degree than the denominator) and find the largest on the positive side, and then say, as far as horizontal asymptotes go, I don't care about the behavior of the function in between those two points, it is only on the ends that it matters.
John
My calculator said it, I believe it, that settles it
A rational function's curve can never cross a horizontal asymptote because
the function is always undefined and approaching infinity or negative infinity as x approaches the x-value of a horizontal asymptote. If the
curve crossed it the function would be defined there.
However a rational function's curve can indeed cross a vertical asymptote
because the curve can cross it at some point, then curve back and approach
it.
The function
has graph:
The curve crosses its horizontal asymptote y = 1 at the point:
Edwin
