On-line Math 21

1.3  Infinite Limits

Exercise 1

lim x® ¥  Ö   x2+2x-1   -x =

Hint

Here the trick is what is called conjugation. Since

(a-b)(a+b) = a2-b2,

whenever we have a difference of square roots,

(Öa-Öb),

we can multiply by the sum of those square roots, top and bottom, and rationalize the numerator,¹

( Öa-Öb) = ( Öa-Öb) ( Öa+Öb) ( Öa+Öb) = ( a-b) ( Öa+Öb) .

Applied to this case,

Ö   x2+2x-1   -x = æ è Ö   x2+2x-1   -x ö ø æ è Ö   x2+2x-1   +x ö ø Ö   x2+2x-1   +x = ( (x2+2x-1)-x2) Ö   x2+2x-1   +x .

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Footnotes:

¹Most of you were taught to always rationalize the denominator of fractions. I never understoond why you should do that. In many, many cases it makes much more sense to rationalize the numerator, since it is numerators that are added together.

Copyright (c) 2000 by David L. Johnson.