If #f(x)=lim_(n->oo)1/(1+nsin^2pix)# , find the value of #f(x)# for all real values of #x#?

1 Answer
Jan 23, 2018

#f(x)=1# when #x# is an integer and #f(x)=0# otherwise.

Explanation:

If #x# is an integer (#x=0, pm 1, pm 2, pm 3,...#), then #pi x# is an integer multiple of #pi# so #sin(pi x)=0#. In this case, #1/(1+n sin^{2}(pi x))=1/(1+0)=1# for all #n#.

If #x# is not an integer, then #sin(pi x) !=0# so #n sin^{2}(pi x)->infty# as #n->infty#. In this case, #1/(1+n sin^{2}(pi x))->0# as #n->infty#.

The answer above follows.