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Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials

I have a question about the best method to find the limit of a function as it approaches infinity.

The limit as x approaches infinity of (x^3+5x)/(2 x^3-x^2+4) is 1/2.

I found this just by taking the largest values of x (X^3/2x^3) and plugging in a number (1). I proceeded under the assumption that if x gets large it will subsume the lesser values of x and the constants, so in a way these lesser values are not necessary.

The book I am using (and other posts on this site), however, provide a different method (i.e., divide the numerator and the denominator by the largest value of x, and then use the limit laws to find the result). This method is obviously much more cumbersome, but I am not sure if any rigor is gained by it.

My question is: Is there a case where the first method will produce a wrong answer? Put another way, will the limit as x approaches infinity of (ax^2+bx+c/dx^2+ex+f) always be equal to the limit as x approaches infinity of (ax^2/dx^2)?

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marked as duplicate by Qiaochu Yuan Sep 11 '11 at 20:23

This question was marked as an exact duplicate of an existing question.

Your method and the textbook method are the exact same, with one fundamental difference: the book provides justification for why the extra values tend to $0$ and can be stricken, whereas yours chops them off without formal justification. Your idea of lesser values being 'unnecessary' is what the book shows. Yes, there is rigor gained, and I simply disagree with you that that's at all cumbersome as far as proofs go. If you're only looking to calculate a limit and you already have knowledge of why this specific rule works then of course you can sidestep actually going through a proof. – anon Sep 11 '11 at 20:16

Your method will always work (assuming $a$ and $b$ are nonzero).

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