Unusual example for limit of composite functions

 Unusual examples of composite limits.

Let g be a function defined on real line by g( x) = x sin 1/x.

 Then limx--> 0g(x)  = 0

 Consider a function f(u) defined on [0 1] by f(u) = 0 when u is irrational and f(u) = 1/q if y is a rational number of the type p/q provided q is the smallest integer in magnitude such  that u = p/q ( we have cancelled common factors). Clearly f(0) = 1.

 Next we observe that limu--> 0 f(u) = 0 but f(0) =1.

In this case the limit of the composite function h =fog does not exist as x tend to 0. It oscillates between 0 and 1 . It takes value  1 when x =  1/nπ  for all integers n. but it takes value  0 when x = 2/nπ . thus in any neighborhood of x =0 it takes both values. So limit of the composite function h(x) does not exist as x tends to 0

2) Consider now G = f og , g(x)  is the constant function  0.

u =g(x) and y =f(u) then limx--> 0 g(x) = 0 and  imu--> 0 f(u) = 0

But limx--> 0 fog(x) = 1  as both f(0) = f(1) =1.

Anil Pedgaonkar