Is there a mathematical symbol meaning "near to" as in "near to infinity?"

Anything a little less vague than <∞?

I think the term you are looking for is 'approaches infnity,' which is '→∞.'

For example, 'x approaches infinity' would be 'x→∞.'
 

There is a symbol that means "approximately equal to." It looks like this: ≈

As to the second part of your question, there is no such thing as "near to infinity."

One can say 'approaching infinity'
with:
lim
x->∞

i guess you could say as x tends to infinity or x→ ∞