Calculus help, please! What does it mean as x of f(x) approaches infinity and the limit approaches infinity?

lim f(x) = L

x--> infinity

As x gets bigger f(x) converges on L

Pick some number N

if x> N

|f(x) - L| is small.

lim f(x) = infinity

x--> a

says something else entirely

This says that f(x) gets to be arbitrarily large as x get close to a.

you could think of the lim x--> infinity as asking if there is a horizontal asymptote

and if the lim x-->a = infinity is saying that there is a vertical asymptote.

this simplification doesn't cover all of the possibilities, but it should give you some intuition.

It means that as you put larger and larger numbers in place of x, the function itself gets larger. It helps to actually try putting in numbers and seeing it for yourself, I feel.

As x gets bigger than any number you can name, so does f(x).