Question

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

Answer 1

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.