This is just not true. The normal limit we have here means a limit would have to exist from both directions and they should be equal. They're not, so the limit doesn't exist.

One-sided limits would be denoted by x -> 5– and x -> 5+ or similar.

PS: in complex analysis, there is no distinction between +infty and -infty, so there it would be correct to say the function has limit infty at 5.