Question

Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement.

(a) lim x-> a [f(x) + g(x)] = infinity
(b) lim x-> a [f(x)g(x)] = infinity if c > 0

I need to prove it using the precise definition of a limit (i.e. NO limit laws).

Answer 1

\(\lim_{x\to a}f(x)=\infty\iff \forall N \exists \delta\) such that \(|x-a|<\delta \implies f(x)>N\)

Answer 2

you can use the same \(\delta\) essentially

Answer 3

or put \(M=N+c\)

Answer 4

See, I've gotten to f(x)+g(x)>M+C-(Epsilon) and I'm kinda clueless.

Answer 5

(Epsilon) = N