What is the limit as x approaches infinity of $(x^2-4)/(2+x-4x^2)$ ?

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Answer 1 · 2014-09-28T22:10:00

$lim_{x to infty}{x^2-4}/{2+x-4x^2}=-1/4$

Let us look at some details.
There are a couple of methods to do this.

Method 1

$lim_{x to infty}{x^2-4}/{2+x-4x^2}$

by dividing the numerator and the denominator by $x^2$ ,
( $x^2$ was chosen to match the leading term in the denominator.)

$=lim_{x to infty}{1-4/x^2}/{2/x^2+1/x-4}={1-0}/{0+0-4}=-1/4$

Method 2

$lim_{x to infty}{x^2-4}/{2+x-4x^2}$

by l'Hopital's Rule ( $infty$ / $infty$ ),

$=lim_{x to infty}{2x}/{1-8x}$

by l'Hopital's Rule ( $infty$ / $infty$ ) again,

$=lim_{x to infty}{2}/{-8}=2/{-8}=-1/4$

I hope that this was helpful.

What is the limit as x approaches infinity of (x^2-4)/(2+x-4x^2)(x^2-4)/(2+x-4x^2)?