Finding the limit of x^(-y) as (x,y)->(1+,∞)

[rman144]

I've been stuck on this problem for quite a while now and could use some assistance:

Find the limit (or prove that it does not exist):

lim{(x,y)->(1^+,oo)} x^(-y)


I've tried switching to polar and end up with y=rsin(@) implying r diverges, which implies cos(@) must tend to zero for x to approach 1, but I'm not certain this actually proves or disproves anything. Honestly, any help would be much appreciated.

 

[Dick]

Try some examples. Take xn=(1+1/n) and yn=n and let n go to infinity. Then xn->1+ and yn->infinity. What's the limit of xn^(-yn)? Then try xn=(1+2/n). Conclusion?

 

[rman144]

Why didn't I think of that? 1/e, 1/e^2, 1/e^3...

Lol, thanks.