Trouble understanding!!

Well there is this theorem and i am having a little trouble understanding a part of it:

-If the function u=g(x) has Uo as the limit at the limit point a (x-->a), but there exists some delta1 neighbourhood of a, such that for 0<Ix-aI<delta1 we have u=g(x)=/=(not equal) Uo, and if the function y=f(u) has b as the limit at the limit point Uo(u-->Uo), then the compound function y=f[g(x)] has b as the limit at the limit point a. That is:

If lim(x-->a) g(x)=Uo ( u=g(x)=/=Uo for 0<Ix-aI<delta1 )

and lim(u-->Uo)f(u)=b then

lim(x-->a)f[g(x)]=b.

The part that i do not fully understand, since i am not able to apply it on problems is the additional requirement on the first part of this theorem If the function u=g(x) has Uo as the limit at the limit point a (x-->a), but there exists some delta1 neighbourhood of a, such that for 0<Ix-aI<delta1 we have u=g(x)=/=(not equal) Uo

My book does not provide any examples at all to ilustrate what happenes if this requirement is not fullfilled, and i am having trouble coming up with any examples that would illustrate this point. SO if someone could post some examples and point out why this requirement is crucial for the theorem to hold, i would really appreciate it.

SO if you can explain this a little, and throw some examples illustrating this i would really appreciate it.