Let and
be functions, both differentiable in a neighborhood of 0, with
and such that
Prove or disprove each the following statements.
-
as
.
-
as
.
- True.
Proof. Sinceas
we know by the definition of
that
Thus, for every
there exists a
such that
So, for
we have
The final line follows since
by hypothesis. Therefore,
Hence,
By definition, we then have
- False.
Considerfor
and
for
. Then, for
,
For
we have
.
Next,
Since
we have
as
. However,
since
does not exist.