Can limits equal $\infty$ or should I say that the limit doesn't exist? So I've seen various questions with the limit 'equal' to $\infty$ or that the limit doesn't exist in a case where the function tends to $\infty$.
For example, the limit of $\sqrt{x}$ as $x$ tends to $\infty$. Is the answer $\infty$ or that the limit doesn't exist?
Obviously the function tends to $\infty$ as $x$ tends to $\infty$ but I don't know what to give as an answer.
I've seen similar questions where the function tends to $\infty$ as $x$ tends to a certain value where the answer has been that the limit doesn't exist. I've also seen where, in a similar situation, the limit has been 'equal' to $\infty$.
So which is the one to use? What's the difference? Thanks!
 A: Depends on the context in which you're working.
If it's with the reals, for example, then such limits simply fail to exist. However, we do sometimes work with the extended reals $[-\infty,+\infty],$ and clearly in that case we can say something like $\lim_{x\to+\infty}x=+\infty.$
A: Usually we say that limit exists when it is finite or finite. In the first case we say that the function converges to $L$, in the second case we say that the function diverges to plus or minus infinity.
We say that the limit doesn’t exist in all the remaining cases, for example periodic functions or other not convergent nor divergent cases.
A: If you could show some examples of the problems you talking about that would help greatly.
For cases where $x \to \infty$, if the limit is increases with out bounds then you would say the limit equals infinity.
$\lim_{x\to\infty}\sqrt{x}=\infty$
$\lim_{x\to\infty} x^2=\infty$
For the instances of the limits that approach a finite value and end up being 'does not exist'.
In these cases the we have to be approaching the same value from the left and the right side of the finite value.  Say you are approaching from the left and heading towards negative infinity and from the right you are approaching positive infinity.  In this case you would say the limit 'does not exist'.  Now if it was approaching positive infinity on both the left and right, you would be able to say the limit equals infinity.
For some examples
$\lim_{x\to2}\frac{x}{x^2-4}\Rightarrow$ Does Not Exist
$\lim_{x\to0}\frac{1}{x^2}=\infty$
