Find the two functions f and g such that the lim x -> 0 f(x) and lim x -> 0 g(x) do not exist, but lim x -> 0 [f(x)+g(x)] does exist.

Question

Even I split

 

lim   [f(x)+g(x)]   into   lim f(x) + lim g(x)

x→0                              x→0         x→0

 

It does nothing as both

 

lim  f(x) AND lim  g(x) do not exist

x→0               x→0

 

Maybe both functions may involved imaginary numbers, eg f(x)=i, g(x)=-i ? If yes, how can i show the steps

Answer 1

Here's an example:

 

Let f(x) = 1/x and g(x) = -1/x

 

lim_x→0f(x) AND lim_x→0g(x) do not exist

 

But, lim_x→0[f(x)+g(x)]  = 0

Answer 2

The limits won't exist if you have a jump discontinuity, such as:

 

            |   3   if x≥0

f(x) =  <

            |  -3  if x<0

 

 

            |   x-3  if x≥0

g(x) =  <

            |   x+3  if x<0

 

But if you add them together, you get a continuous function:

 

                   |  3+(x-3) = x  for x≥0

[f+g](x)  =  <

                   |  -3+(x+3) = x  for x<0

 

[f+g](x) = x  which has a limit of 0 as x→0