Proving the Limit of a Reciprocal Function

[Bigo75]

Homework Statement

Prove: If the Lim as x goes to a of F(x)=infinity, then lim as x goes to a of 1/F(x)=0

Homework Equations

We have only gone over limits and continuity using delta epsilion proofs.

 

[ircdan]

what have you tried? this is straightforward in the sense that it follows from the definitions

show us what you have tried

 

[Bigo75]

Since the limit is equal to infinity I took an M greater than zero such that X>M implies |F(x)-L| is less than epsilon but I get confused with the one over F(x)=0

 

[Bigo75]

am I on the right track? I am really confused!

 

[ircdan]

Since the limit is equal to infinity I took an M greater than zero such that X>M implies |F(x)-L| is less than epsilon but I get confused with the one over F(x)=0

this is wrong, lim x->a F(x) = inf means for all M > 0 there is a d > 0 s.t. |x-a| < d implies |F(x)| > M. (you can actually drop the absolute value sign here but i'll leave it)

 

[Bigo75]

ok I understand this definition but how does that help me get lim as x goes to a of 1/f(x)=0? This definition will help me prove another one I was having trouble with. thank you.