Use L''hopital''s rule
since lim X-->0 1-cos(x)/1-cos(4x) is in the indeterminate form 0/0 when we apply the limt
so by l''hoptital''s rule differentiate the numerator and denominatior
so we get lim x-->0 sin(x)/4sin(4x) again in the indeterminate form when we apply limit so again use l''hopitals'' rule
lim x-->0 cos(x)/16cos(4x) now if u apply the limit we get it as 1/16
so lim x--->0 1-cos(x)/1-cos(4x) = 1/16