Is this solution to a limit equation correct?

[nedfin]

Homework Statement

I ran into this equation online and an interested in why my answer might be wrong. What if anything is wrong with my logic.

Relevant Equations

1-(1/((x-y)^x))
lim(x,y) -> (inf,0)

Why is this not equivalent to

1 - inf^-inf,

Or 1 - infinitesimal ,

Or 1 ?

My answer was 1, which I told is incorrect.

 

[nuuskur]

$$\lim _{(x,y)\to (\infty,0)} \frac{1}{(x-y)^{x}} = 0$$
because $(x-y)^x \to \infty$ as $(x,y)\to (\infty,0)$. It is not equivalent to the first two items. $\infty ^{-\infty}$ is not a thing and neither is "infinitesimal". The initial limit is $1$ as you say.

 

[nedfin]

$$\lim _{(x,y)\to (\infty,0)} \frac{1}{(x-y)^{x}} = 0$$
because $(x-y)^x \to \infty$ as $(x,y)\to (\infty,0)$. It is not equivalent to the first two items. $\infty ^{-\infty}$ is not a thing and neither is "infinitesimal". The initial limit is $1$ as you say.

Thanks

 

[berkeman]

Homework Statement:: I ran into this equation online and an interested in why my answer might be wrong. What if anything is wrong with my logic.
Relevant Equations:: 1-(1/((x-y)^x))
lim(x,y) -> (inf,0)

Why is this not equivalent to

1 - inf^-inf,

Or 1 - infinitesimal ,

Or 1 ?

My answer was 1, which I told is incorrect.

Welcome to PF. I'm glad that @nuuskur was able to help you.

BTW, please consider learning LaTeX to post equations in discussion forums. You can find a "LaTeX Guide" link at the bottom of the Edit Window. It makes math equations *much* easier to read (as you can see by nuuskru's post).