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Having some trouble expressing the following:

Find $\displaystyle \lim_{x\to\infty} \frac{2-3x+4x^2}{7 + 12x + 3x^2}$, by first expressing the limit as $\displaystyle \lim_{x\to0} f(x)$, for some real function $\displaystyle f$

My initial thoughts would be to switch the numerator and the denominator and then evaluate at $\displaystyle x\to0$, which would leave an answer of $\displaystyle \frac{7}{2}$, but this seems too easy and is thus more than likely incorrect.

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2 Answers 2

up vote 6 down vote accepted

First write $$ {2-3x+4x^2\over 7+12x+3x^2}= {{2\over x^2}-{3\over x}+4\over {7\over x^2}+{12\over x}+3}. $$ Then the limit is expressed by (setting $u=1/x$ in the above) $$ \lim_{u\rightarrow0}{{{2u^2}-{3u}+4\over {7u^2}+{12u}+3}} . $$

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Aha! Because 1/x tends to 0! So as x tends to infinity 1/x = u tends to 0, thanks David. –  Billy Ray Valentine Apr 10 '12 at 14:11
    
@BillyRayValentine You're welcome. (Ignore my last (deleted) comment). –  David Mitra Apr 10 '12 at 14:15

Changing variables is inefficient. Rather, simply employ the useful Principle of Domination to deduce the that limit of a quotient is the limit of the quotient of the "dominant" terms, namely factor out the dominant terms from each sum as follows (where limits are as $\rm\:x\to \infty)$ $$\rm\ \frac{f}g\to a,\ \ \frac{f_j}f \to 0,\ \ \frac{g_j}g \to 0\ \ \Rightarrow\ \ \frac{f + f_1 +\cdots f_n}{g + g_1\cdots g_k} \ =\ \frac{f}g\: {\frac{1 + \dfrac{f_1}f+\cdots+\dfrac{f_n}f}{1 + \dfrac{g_1}g +\cdots + \dfrac{g_k}g}}\:\!\to\: a $$

Therefore $\rm\ \ \dfrac{x}{x^2} = \dfrac{1}{x}\to 0,\ \ \dfrac{1}{x^2}\to 0\ \ \Rightarrow\ \ \dfrac{ax^2+bx+c}{dx^2+ex+f}\ =\ \dfrac{a + \dfrac{b}x + \dfrac{c}{x^2}}{d + \dfrac{e}x + \dfrac{f}{x^2}}\to\: \dfrac{a}d $

Should you go on to study the algebraic approach to limits and asymptotics via valuation theory you will find that an analogous principle of domination plays a key role.

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Thanks Bill, this is beyond what I need for my current course but is useful to know. –  Billy Ray Valentine Apr 11 '12 at 8:47
    
@Billy The remark in the final paragraph may be beyond what you need to know, but the rest is essential for efficiently calculating limits. Without using that principle, calculation of many limits will prove much more difficult than need be. –  Bill Dubuque Apr 11 '12 at 15:26

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