Take 15 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
If you have more questions, feel to ask then on the forum.
Question
By using Taylor-expansions, find the following limits:
a)
b)
c)
where
are three strictly positive real numbers.
a) Find a Taylor-expansion of order
of application
by using those of applications
and
then find a Taylor-expansion of order
of the numerator.
Do the same with the denominator.
b) By using a Taylor-expansion of
put
in the form
then use the Taylor-expansion of application
Infer a Taylor-expansion of order
of the numerator by using the Taylor-expansion of exponential application.
c) Set
Use the equality
and the Taylor-expansion of exponential application to put the numerator in the form
Infer the limit of
then the one of
by using the continuity of the exponential application.
a)
(the
is even, so no term in
We then have:
Let us now determine a Taylor-expansion of
of order 3.
We infer:
We infer that:
b)
We then have:
and we conclude:
c) We set
and
goes towards
when
goes to
therefore:
We infer that :
We infer:
and:
