Friday 25 March 2011

The Sharp Edge Between Infinity and "Finity"

After my problem on the steel rail got me in trouble, I have decided to stick to a purely theoretical problem that has no messy reality features to get in the way...

I recently came across two limit problems that got me to thinking about how narrow the line between infinity and the finite can be....

Try the following two limits.....

Both can be simplified by the old "multiply by one" trick ... letting the "one" in this case be the conjugate of the binomial ...For the first, it works like this...
Ok, sorry, I got those last two images reversed... the middle one is the simplest........
Which leads to the conclusion that the limit is 1/2... Ok, not terribly difficult, so what...???

 But if we do the same sort of thing with the second, we get :

Yikes, now the Limit has shot off to infinity... The difference between a one and a two for that leading coefficient has made a huge difference. At first we might think that we can find some number a such that :

has a finite limite between 1/2 and infinity, But when you try it.... if a is greater than 1, the limit is infinity, and if a is less than one, the limit is negative infinity, and in between, balanced on a razor's edge is a=1, which has a limit of 1/2...
Just goes to show that 1/2 is the average of Plus and minus infinity,

1 comment:

Dave L. Renfro said...

Another way to see this is to recall the linear approximation sqrt(1 + u) = 1 + u/2 for |u| near zero (the error is roughly x^2/8).

For positive values of a and x we have

sqrt(ax^2 + ax)

= sqrt[(ax^2)*(1 + 1/x)]

= x*sqrt(a)*sqrt(1 + 1/x).

Thus, for large values of x this is approximately

x*sqrt(a)*[1 + 1/(2x)],

with an error of approximately

x*sqrt(a)*(1/x)^2 / 8

= sqrt(a) / (8x)

Thus, for large values of x we have approximately

sqrt(ax^2 + ax) = x*sqrt(a) + sqrt(a)

and, even with the addition of sqrt(a), the result you got is now transparent.

By the way, I made a comment in your March 20 entry that others (who might not have thought to check back there) might be interested in.