Need help with the following proof:
prove that if lim x->c 1/f(x)= 0 then
lim x->c f(x) does not exist.I think i need to use the delta epsilon definition i am not sure how to set it up.
Proof by contradiction. Suppose that the limit exists, and that $\displaystyle \lim_{x\to c}f(x) = L$.
We know that $\displaystyle \lim_{x\to c}1/f(x) = 0$. So for each $\displaystyle \varepsilon>0$ there exists $\displaystyle \delta>0$ such that $\displaystyle |1/f(x)|<\varepsilon$ whenever $\displaystyle |x-c|<\delta$.
Choose $\displaystyle \varepsilon< 1/(|L|+1)$. Then $\displaystyle |x-c|<\delta\ \Rightarrow\ |1/f(x)|< 1/(|L|+1)\ \Rightarrow\ |f(x)|> |L|+1$.
But that means that whenever x is sufficiently close to c, f(x) differs from L by at least 1. That contradicts the assumption that $\displaystyle f(x)\to L$ as $\displaystyle x\to c$. (I'll leave it to you to make that last sentence rigorous in terms of epsilons and deltas.)