DNE vs Toward Infinity

[Jason76]

Sometimes a limit will march toward positive or negative infinity, while sometimes DNE. So how can we tell the difference (without using a graph, or by using algebra)? Or is DNE and going toward positive and negative infinity the same thing (or sort of the same)?

 

[daon2]

"Marching off to +infinity" is an example of "DNE." It is a specific kind that describes the behavior of divergence. Other limits such as of x*sin(x) does not go off to one infinity or the other, so a simple does not exist is appropriate. In more advanced texts, they may treat + and -infinity as a number (part of the "extended reals"), and so the limit of, for example, f(x)=x^2 would exist.