I've been tackling a few analysis problems today and have once again hit a bit of a block.
I need to either prove or find a counterexample to the following two propostions:
Noting here that I am saying a limit exists if it converges in In other words, if then it DOES exist. I know some take an infinite limit as not existing but I am taking it as existing for the purpose of this problem. A non-existent limit would be one like sin(x) where it just sort of oscillates or where the left limit and right limit are not equal.
So, any pointers here? I am pretty sure that both are true because I've had no luck finding a counter example. In a similar problem to this I had earlier (which was effectively the reverse) I was able to employ the Mean Value Theorem.
But basically I am completely stuck.
I thought about saying the following:
We know that if a function is differentiable then it must be continuous. We know that a function is continuous if and only if it has a limit at every point. Hence, if as then f(x) is continuous as
Thus f(x) has a limit as x
But I'm pretty sure I'm mis-remembering something with that claim in italics and thus not making any sense at all. Any pointers would be greatly appreciated.