I have been taught that if the left and right hand limits of a function are not equal, then there is not a limit for that function. However I have also seen functions which tend to positive and negative infinity depending on from which direction you evaluate it. So was the rule I was taught not strictly correct, rather if a function has left and right limits that are not of equal magnitude then there exists no limit, but if they are of the same magnitude but opposite sign, we can say there exists a limit which is plus/minus the magnitude?
If the left and right hand limits of a function have equal magnitude but opposite sign, does there exist a limit for that function
limits
Best Answer
Absolutely not. If there is a nonzero left limit $L$ and a right limit $R = -L$, then there's no "un-directional" limit. Example
$$ f(x) = \begin{cases} -1 & x \le 0 \\ 1 & x > 0 \end{cases} $$ has $L = -1, R = 1$, but does not have a limit at $0$.
In the case where $L$ and $R$ both are zero, I guess one could say that $L = -R$ and the limit still exists, but that's hardly an interesting case. :)
In the case that the left limit is infinity, and the right limit is $-\infty$, there's also no limit at the point. An example is $$ f(x) = \begin{cases} -1/x & x \ne 0 \\ 1 & x = 0 \end{cases} $$ This function has left limit $\infty$ at $0$, right limit $ -\infty$ at $0$, but has no limit at $0$.