Limits with the precise definition of a limit

[newschoolgg]

Homework Statement

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

Homework Equations

The limit laws would seem relevant, but the f(x) limit goes to infinity.

The Attempt at a Solution

I'm completely lost on how to start this problem. How would I prove something like the limit laws using the precise definition of a limit when one of the limits don't exist?

 

[vela]

When you say that
$$\lim_{x \to a} f(x) = \infty,$$ it's not exactly the same as saying the limit doesn't exist. For example, the limit
$$\lim_{x \to 0} \sin \frac{1}{x}$$ doesn't exist, but you wouldn't say it's equal to infinity either. So what precisely does it mean when you write a limit equals infinity?

 

[Mark44]

Homework Statement

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

Homework Equations

The limit laws would seem relevant, but the f(x) limit goes to infinity.

The Attempt at a Solution

I'm completely lost on how to start this problem. How would I prove something like the limit laws using the precise definition of a limit when one of the limits don't exist?

Your textbook should have the precise definition of a limit when the function is unbounded. It doesn't use $\delta$ and $\epsilon$ as the normal limit does, but instead uses M and $\delta$, where M is a large number.