Question

exercise 3.1.8: find example functions f and g such that the limit of neither f (x) nor g(x) exists as x!0, but such that the limit of f (x) g(x) exists as x!0.

Answer 1

functions f and g such that the limit of neither f (x) nor g(x) exists as x!0.

let,

f(x)  =  1    at x=Q

also, f(x)  = -1  at x=Q'

now, g(x)  = -1  at x = Q

also, g(x) = 1 at x = Q'

Here,

$\lim_{n \to \infty} f(x)$      and $\lim_{n \to \infty} g(x)$   does not exist.

but,    f(x) + g(x)  =  (f+g)(x) = 0 at x = Q and also at x = Q'

⇒  (f+g)(x) = 0 ∀ x∈IR

⇒   $\lim_{n \to \infty} (f+g)(x)$  = 0 exist.

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