How do you identify the oblique asymptote of #( 2x^3 + x^2) / (2x^2 - 3x + 3)#?

1 Answer
Tony B
Nov 4, 2015

See explanation.

Explanation:

You have to consider the extreme cases for asymptotes and these will vary according to the equation concerned. Usually it involves #lim_(x-.oo)# but it may also involve#lim_(x->"some number")#. In your case I am assuming we are not looking at the behaviour near #x=0#.

#color(brown)("Consider the numerator:")#
#2x^3# grows much faster than #x^2#. Consequently #2x^3# 'wins' as the dominant factor as #x -> oo#

#color(brown)("Consider the denominator:")#
For the same reason as above, #2x^2# 'wins' as the dominant factor as #x -> oo#.

#color(brown)("Put together:")#
Thus as #x -> oo# we have #lim_(x-> oo) (2x^3+x^2)/(2x^3-3x+3) =(2x^3)/(2x^2) = x#

Note that #x# can be positive or negative.

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