# Thread: Multivariable Epsilon Delta Proof

1. ## Multivariable Epsilon Delta Proof

Hi MHF,

I'm looking for a little help with an Epsilon Delta proof. I'm taking a deferred exam 6 months after the course ended, so I really need help here.

The Question: Let f be a function defined on a set S in R^n and suppose Q is a limit point of S. If $\lim_{P \to Q} f(P) = 2$, prove from first principles that $\lim_{P \to Q} [1/f(P)] = 1/2$

My Attempt By definition, we know that $|P-Q| < \delta ==> |f(P) - 2| < \epsilon$. Suppose $\epsilon ' = 1/\epsilon$

This is really all I have and understand and is probably wrong. I apologize in advance that I may have many questions in the following days as I'm slowly realize I barely understand anything from this course any more and I am truly trying but this is making no sense to me and I have no reference (Textbook was rented).

Any help would be greatly Appreciated.

2. Originally Posted by rtplol
Hi MHF,

I'm looking for a little help with an Epsilon Delta proof. I'm taking a deferred exam 6 months after the course ended, so I really need help here.

The Question: Let f be a function defined on a set S in R^n and suppose Q is a limit point of S. If $\lim_{P \to Q} f(P) = 2$, prove from first principles that $\lim_{P \to Q} [1/f(P)] = 1/2$

My Attempt By definition, we know that $|P-Q| < \delta ==> |f(P) - 2| < \epsilon$. Suppose $\epsilon ' = 1/\epsilon$

This is really all I have and understand and is probably wrong. I apologize in advance that I may have many questions in the following days as I'm slowly realize I barely understand anything from this course any more and I am truly trying but this is making no sense to me and I have no reference (Textbook was rented).

Any help would be greatly Appreciated.
First of all 1/epsilon is a very huge number!

3. have you given any thought as to how one might re-write |1/f(P) - 1/2| in terms of something involving f(P) - 2 (which you know you can make "smaller than epsilon")?

4. And also...
$\lim_{P\to Q} f(P)=2$ hence in small $\delta$ neighborhood the function $f(P)$ is bounded. ( $2-\epsilon)