Oblique asymptotes - What am I doing wrong?

[jachyra]

f(x) = (2x^3 + 4x^2 - x + 1) / (-x^2 - x + 2)

The limit of this function as x approaches infinity is the oblique asymptote f(x) = -2x - 2
This can be verified by performing long division with the two polynomials to get:

f(x) = -2x -2 + (x +5)/(-x^2 - x + 2)
as x -> infinity, the term (x +5)/(-x^2 - x + 2) -> zero

Now my question is why does the following not work:

f(x) = (2x^3 + 4x^2 - x + 1)/(-x^2 - x + 2) * (1/x^2)/(1/x^2)

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

lim f(x) as x -> infinity should then be -2x - 4 because all other terms approach zero as x gets larger and larger right?

why is this wrong? I am so confused... I thought the answers should have been the same!

 

[mathman]

The denominator in your expression makes a contribution to the constant term.
1/(-1-1/x)~-1+1/x. Therefore f(x)~(2x+4)(-1+1/x)~(-2x-4)+(2+4/x)~-2x-2

 

[tacman]

There could be a few reasons why your approach did not yield the same result. One possibility is that there may have been a mistake in your long division or simplification of the fractions. Another possibility is that the algebraic manipulation you performed may have changed the original function in some way. It is important to keep in mind that when manipulating equations, we need to ensure that the resulting equation is equivalent to the original one.

Additionally, it is important to note that when finding the oblique asymptote, we are looking at the behavior of the function as x approaches infinity, not just at a specific value of x. So even if your approach gave the same result for a particular value of x, it may not be accurate for the overall behavior of the function.

In general, it is always a good idea to double check your work and make sure you are following the correct steps when solving math problems. If you are still unsure about why your approach did not work, it may be helpful to ask a teacher or tutor for clarification.