Proving the Limit of f(x) as x->0 and Lim[e^n-(1+1/n)^n^2]

[azatkgz]

f(x)=1,xeQ
f(x)=0,x in not eQ

It's easy to understand that this function has no limit as x tends to 0,but how we can prove it with mathematical formulas.

And what's the lim[e^n-(1+1/n)^n^2]

 

[quasar987]

Prove it with the definition of limit.

Can you find an epsilon such that no delta-nbhd of 0 is mapped entirely in an epsilon-nbhd of f(0)=1?

Or use the characterisation in terms of sequences. The limit is f(0) iff for every sequence converginf to 0, the image sequence converges to f(0). Can you find a sequence such that the image sequence does not converge to f(0)?

 

[azatkgz]

quasar987,
I could not solve with sequences.Can u please post the solution.

 

[HallsofIvy]

So far you haven't shown that you have tried anything at all! What have you tried?

Have you thought about what f(x) is if x is close to 0 and rational?
What f(x) is if x is close to 0 and irrational?
COULD f(x) be "close" to some limit if x is any number close to 0?