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 $\displaystyle f(x)=\tfrac2{5x}$ grows without bound as $\displaystyle x$ tends to zero from above, and this is what the notational shorthand $\displaystyle \lim_{x \to 0^+} \tfrac2{5x}=\infty$. Note that "infinity" is not involved at all because real analysis has no "infinity".
Also note that $\displaystyle \lim_{x \to 0} \tfrac2{5x}$ does not exist because $\displaystyle \lim_{x \to 0^+} \tfrac2{5x} \ne \lim_{x \to 0^-} \tfrac2{5x}$. 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 $\displaystyle -\infty$", 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).