How do you find the slant asymptote of # ( x^4 + 1 ) / ( x^2 + 2 )#?

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Answer 1 · 2016-01-15T21:18:33

This rational function is asymptotic to a parabola, not a line.

It has no slant asymptote.

The degree of the numerator is #4# and the degree of the numerator is #2#.

As a result this rational function is asymptotic to a parabola, not a line.

More explicitly:

#f(x) = (x^4+1)/(x^2+2)#

#=(x^4+2x^2-2x^2-4+5)/(x^2+2)#

#=(x^2(x^2+2)-2(x^2+2)+5)/(x^2+2)#

#=x^2-2 + 5/(x^2+2)#

So as #x->+-oo# we find #(f(x) - (x^2-2)) -> 0#

That is #f(x)# is asymptotic to #x^2-2#