OK, I have the equation (4+x^4)/(x^2-x^4) I need to find all of the vertical and horizontal asymptotes...
I would like to know step by step how I find them all... thanks for your help!
I would like to know step by step how I find them all... thanks for your help!
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Vertical asymptotes of a rational function happen when the denominator is 0. (...but only if the numerator is nonzero. If both are zero at a particular value x=c, then there's a common factor of (x-c) in the numerator and denominator that you must remove to find out what really happens near x=c.)
Apply that here; x^2 - x^4 = x²(1 - x²), which can be factored further to x²(1 - x)(1 + x). So the denominator is 0 if x is 0, 1 or -1. The numerator is never 0, so there's no problem with 0/0. You have vertical asymptotes x=0, x=1 and x=-1.
Horizontal asymptotes happen, only when the rational function approaches a constant value as x tends to either +infinity or -infinity. That can only happen when the numerator and denominator have the same degree. The limit is the ratio of the leading terms (in standard form).
Apply that here. Putting the polynomials in standard form, top and bottom:
y = (x^4 + 4) / (-x^4 + x^2)
Those are both degree 4, so the horizontal asymptote exists. The ratio of leading terms gives:
y ≈ (x^4) / (-x^4) = -1 ... for very large positive or negative values of x. The horizontal asymptote is
y = -1.
Take a look at Wolfram Alpha for a plot of the function:
http://www.wolframalpha.com/input/?i=y+%…
You can graphically see that the curves go vertical near x=-1, 0 and 1, and kinda see that the sides flatten out to approach y=-1. (The limit as x->infinity is at the bottom, stated as -1, so the -1 is exact.)
Apply that here; x^2 - x^4 = x²(1 - x²), which can be factored further to x²(1 - x)(1 + x). So the denominator is 0 if x is 0, 1 or -1. The numerator is never 0, so there's no problem with 0/0. You have vertical asymptotes x=0, x=1 and x=-1.
Horizontal asymptotes happen, only when the rational function approaches a constant value as x tends to either +infinity or -infinity. That can only happen when the numerator and denominator have the same degree. The limit is the ratio of the leading terms (in standard form).
Apply that here. Putting the polynomials in standard form, top and bottom:
y = (x^4 + 4) / (-x^4 + x^2)
Those are both degree 4, so the horizontal asymptote exists. The ratio of leading terms gives:
y ≈ (x^4) / (-x^4) = -1 ... for very large positive or negative values of x. The horizontal asymptote is
y = -1.
Take a look at Wolfram Alpha for a plot of the function:
http://www.wolframalpha.com/input/?i=y+%…
You can graphically see that the curves go vertical near x=-1, 0 and 1, and kinda see that the sides flatten out to approach y=-1. (The limit as x->infinity is at the bottom, stated as -1, so the -1 is exact.)