Hi I want to ask a question: what is the limit of let's say 2/5x. Isn't the limit infinity? That's what I find reasonable, however in a limit calculator it says the limit doesn't exist because it diverges. What does it mean? Thanks.
A limit (if it exists) is a real (finite) number. Therefore no limit is "infinity". We can say that the function grows without bound as tends to zero from above, and this is what the notational shorthand . Note that "infinity" is not involved at all because real analysis has no "infinity".
Also note that does not exist because . The inequality is there for two reasons:
- neither limit expression has a finite value at all, so those values cannot be equal;
- the left-sided limit expression grows in a negative direction (informally "goes to ", so even in the loosest sense, saying that the limits are "infinity" and "negative infinity" respectively, they are not equal.
Somehow Archie got it right although I really did forget to include some information. Yes, I asked the question 2/ 5x is approaching where when x is approaching to 0. So if I understand correctly, the limit of 1/x is undefined because the limit diverges when you are approaching from the negative side as opposite to approaching from the positive side. And if I may please explain to me how to write a mathematical expressions here.
There is no one-sided limit from either side because the function diverges, it grows without bound.
There is no two-sided because the one-sided limits don't exist (and are therefore not equal). The definition of the two-sided limit is that it is equal to the one-sided limits when both exist and are equal).