Give an example of a sequence (a_n) and a function f(x) for which a_n = f(n) for every n=1, such that the limit as n approaches infinity of a_n converges but the limit of f(x) as n approaches infinity diverges.

Question

Give an example of a sequence (a_n) and a function f(x) for which a_n = f(n) for every n>=1, such that the limit as n approaches infinity of a_n converges but the limit of f(x) as n approaches infinity diverges.

amistre64 · Answer

maybe a trig function?

anonymous · Answer

Would sin(pi*x) work?

anonymous · Answer

So it would converge to sin(pi) or 0...

anonymous · Answer

But the function itself oscillates so it's limit diverges?

perl · Answer

sin(Pi*n) , for n=0,1,2,3... 
converges to 0. so that works . but sin(pi*x) oscilattes

anonymous · Answer

yes so it satisfies the statement?

anonymous · Answer

lim of sin(pi*x) oscillates so it diverges?

perl · Answer

correct

perl · Answer

the limit does not exist

anonymous · Answer

Sounds great :)

anonymous · Answer

Wait so you wouldn't say a limit diverges but just does not exist?

anonymous · Answer

Okay I got it, thank you :)

perl · Answer

diverges means the opposite of convergence, does not converge

perl · Answer

and thus the limit does not exist. a limit is a finite number