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.

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