Asymptotes - Help? f(x)=x²-5x+7/(x-2)?
f(x)=x²-5x+7/(x-2)
It asks me to give the vertical, horizontal and slant asymptotes if there are any.
This is what I have so far:
vertical asymptote: x cannot equal 2
I am now trying to find the slant asymptote by synthetic (long division) because x²-5x+7 does not factor out.
I am really stuck so may someone show me how to do this problem? I still need to find the slant and horizontal asymptotes.
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Verified answer
There aren't any horizontal asymptotes, as I understand the term. You might note that the range does not include -3 < f(x) < 1.
As for the slant asymptote... Rewrite the expression like this:
(x²-5x+6+1)/(x-2)
=((x-3)(x-2) + 1)/(x-2)
=(x-3)(x-2)/(x-2) + 1/(x-2)
=x-3 + 1/(x-2)
As x gets really, really positive or really, really negative, the 1/(x-2) term gets really close to zero, and you are left with x-3. So the slant asymptote is x-3 in both the positive and negative directions. That is, the graph of f(x) approximates y=x-3.
Cheers,
Nicholas