Let f(x) be the greatest integer less than or equal to x.

Compute lim x->0 xf(1/x). Is the following solution correct.

f(1/x)=[1/x]

lim x->0 xf(1/x) = lim x->0 x[x]

Note that:

(1/x)-1 =< [1/x] =< 1/x

multiply by x:

1-x =< x[1/x] =< 1

since the limit x->0 (1-x)=1 and the limit x->0 1= 1 then

by squeeze theorem the lim x->0 x[1/x] = 1.