Question

Show by means of an example that lim x-> a [f(x) + g(x)] may exist even though neither lim x-> a f(x) and lim x -> a g(x) exists.

Answer 1

To evaluate the limit, we substitute zero to function 1/x resulting to infinity. Hence,
limit from left = -∞ 
limit from right = ∞ 
Thus,
f(x)* g(x) = (1/x) *(1/x) = 1/x² 

as x ->0: 
lim from left = ∞ 
lim from right = ∞ 
lim = ∞

Limit thus doesn't exist