1. ## Functions without limits

What does it mean to say that f(3) exists but that there is no limit to f(x) as x approaches 3?

2. It means that the limit is dependent upon the path you take.

3. Originally Posted by Random Variable
It means that the limit is dependent upon the path you take.
Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about $f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}$

4. Originally Posted by Drexel28
Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about $f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}$
So there's no limit on that one because 1/|x-3| converges on infinitesimality?

5. Originally Posted by satx
So there's no limit on that one because 1/|x-3| converges on infinitesimality?
Did you just say infinitesimality? haha. It diverges...thus it does not exist.

6. Originally Posted by Drexel28
Did you just say infinitesimality? haha.
Don't hate; it's a perfectly cromulent word

7. Originally Posted by satx
What does it mean to say that f(3) exists but that there is no limit to f(x) as x approaches 3?
$f(x) = \left\{ {\begin{array}{rl} {x~~,} & {x \geqslant 3} \\ {x - 1,} & {x < 3} \\ \end{array} } \right.$

8. Originally Posted by Drexel28
Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about $f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}$
Obviously if the function increases (or decreases) without bound then the limit doesn't exist. But what's another way?

9. What about a function that oscillates wildly near a point?

10. Originally Posted by Random Variable
What about a function that oscillates wildly near a point?
Indeed another. What about the function not even existing on an open ball around the point while we're going here.

11. Originally Posted by Drexel28
Indeed another. What about the function not even existing on an open ball around the point while we're going here.
Does that count?

If I remember correctly, in real analysis we proved that $\lim_{x \to 0} \sin(1/x)$ does not exist by finding a sequence $(s_n)$ that converges to some value c ( $s_{n} \ne c$) but where $f(s_n)$ doesn't converge (or something like that).

12. Originally Posted by Random Variable
Does that count?

If I remember correctly, in real analysis we proved that $\lim_{x \to 0} \sin(1/x)$ does not exist by finding a sequence $(s_n)$ that converges to some value c ( $s_{n} \ne c$) but where $f(s_n)$ doesn't converge (or something like that).
Taking $a_n=\frac{1}{2\pi n+\frac{\pi}{2}},b_n=\frac{1}{2\pi n}$