Differantiation proof question

[transgalactic]

here is the question and how i tried to solve it??

http://img510.imageshack.us/img510/441/60031828xt4.th.gif

 

[Gib Z]

Well good, you got quite far. Look at where your limits got you to. You can take out the $$\phi (x)$$ out from the limits now can't you? Your remaining limit, what does that approach from the left? How about from the right?

 

[transgalactic]

i can't take out
$$\phi (x)$$

out of the limit .
its not a constant
??

 

[Gib Z]

Well really it doesn't matter whether you leave it in there or not, its still easy to see that we get different limits from the left and right just by considering the other things in the limit. But we can take it out of the limit because we can regard it as a constant with respect to the limiting variable.

 

[transgalactic]

i can't see
how we get different results

there is another function inside
i don't know how to get a result
??

 

[Gib Z]

Consider

$$\lim_{x\to 0} \frac{ |x| }{x}$$. What is it from the left? How about right? So it does exist?