Functions without limits

**satx** · February 26th 2010, 02:57 PM

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

**Random Variable** · February 26th 2010, 03:08 PM

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

**Drexel28** · February 26th 2010, 03:11 PM

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 $f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}$

**satx** · February 26th 2010, 03:15 PM

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?

**Drexel28** · February 26th 2010, 03:18 PM

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.

**satx** · February 26th 2010, 03:22 PM

Originally Posted by Drexel28

Did you just say infinitesimality? haha.

Don't hate; it's a perfectly cromulent word

**Plato** · February 26th 2010, 03:26 PM

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?

$f(x) = \left\{ {\begin{array}{rl} {x~~,} & {x \geqslant 3} \\ {x - 1,} & {x < 3} \\ \end{array} } \right.$

**Random Variable** · February 26th 2010, 03:27 PM

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?

**Random Variable** · February 26th 2010, 03:28 PM

What about a function that oscillates wildly near a point?

**Drexel28** · February 26th 2010, 03:31 PM

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.

**Random Variable** · February 26th 2010, 03:43 PM

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).

**Drexel28** · February 26th 2010, 03:52 PM

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 $a_n=\frac{1}{2\pi n+\frac{\pi}{2}},b_n=\frac{1}{2\pi n}$

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