Need help proving limit does not exist!

mathie_geek

I got this after: [itex](|x|^{a+(bc/d)-c}) /{2}[/itex]
expanding the exponents I have [itex]\displaystyle \frac{acd+bc^2-c^2d}{cd}[/itex]
and given [itex]\displaystyle \frac{ad+bc-cd}{cd}<=0[/itex] which is 1/c of the exponent..
So then |x| has an exponent negative so limit d.n.e right? cause limit of 1/|x| as x->0 d.n.e

SammyS

Actually, that's [itex]\displaystyle \frac{ad+bc-cd}{d}\le 0[/itex]

What if [itex]\displaystyle \frac{ad+bc-cd}{d}=0\ ?[/itex]


Now, maybe you can adjust the path so that the exponent is always zero.

mathie_geek

Oh, I see. Since [itex]\displaystyle \frac{ad+bc-cd}{d}> \displaystyle \frac{ad+bc-cd}{cd}[/itex] then does that mean it HAS to = 0? If it's 0 then lim is 1/2

SammyS

Let's go back to the exponent:

[itex]\displaystyle
\frac{acd+bc^2-c^2d}{cd}=\frac{c(ad+bc-cd)}{cd}=\frac{ad+bc-cd}{d}\ .
[/itex]

I have no idea as to why you multiplied that by 1/c .


Unless you meant to say that "a,b,c,d are positive integers numbers" (emphasizing integers, rather than numbers), the following inequality

[itex]\displaystyle
\frac{ad+bc-cd}{d}>\frac{ad+bc-cd}{cd} \ \ \text{ is not true.}[/itex]

The number, c, might very well be less than 1 if a, b, c, and d are not required to be integers.

All that I said was, if a/c + b/d ≤ 1, then [itex]\displaystyle \frac{ad+bc-cd}{d}\le 0[/itex].

If you choose the case of it being an equality, (choose the = sign), then the limit is defined, with said limit being 1/2.

However, that's far from being the general case for this expression.