For each positive integer and and all real define
Prove that
Proof. First, we have
for all . Hence,
On the other hand,
Hence,
For each positive integer and and all real define
Prove that
Proof. First, we have
for all . Hence,
On the other hand,
Hence,
Consider the function
and .
where is a polynomial in .
by Theorem 7.11 (page 301 of Apostol) since implies as well
So, indeed the formula is valid in the case . Assume then that the formula holds for some positive integer . We want to show this implies the formula holds for the case .
But then the leading term
is still a polynomial in since the derivative of a polynomial in is still a polynomial in , and so is the sum of two polynomials in . Therefore, we have that the formula holds for the case ; hence, it holds for all positive integers
So, indeed and the statement is true for the case . Assume then that for some positive integer . Then, we use the limit definition of the derivative again to compute the derivative ,
This follow since is still a polynomial in , and by the definition of for . But then, by part (a) we know
Therefore,
Thus, the formula holds for the case , and hence, for all positive integers
Find the value of the following limit.
We apply L’Hopital’s rule (twice) to compute the limit (Noting that, ),
Since this limit exists, the application of L’Hopital’s rule was justified.
Find the value of the following limit.
We compute as follows,
Since this limit exists, the application of L’Hopital’s rule was justified.
Find the value of the following limit.
First, since we can multiply by this inside the limit without changing the value,
Now we can apply L’Hopital’s rule to compute the limit,
Since this limit exists, the application of L’Hopital’s rule was justified.
Find the value of the following limit.
Applying L’Hopital’s rule we have,
(We used that from this exercise, Section 7.11, Exercise #1). Since this limit exists, the application of L’Hopital’s rule was justified.
Find the value of the following limit.
We apply L’Hopital’s rule three times,
Since this final limit exists, the chain of applications of L’Hopital’s was justified.
Find the value of the following limit.
We will apply L’Hopital’s rule three times,
Since this final limit exists, the chain of applications of L’Hopital’s was justified.
Find the value of the following limit.
We apply L’Hopital’s rule,
Since this limit exists, the application of L’Hopital’s was justified.
Find the value of the following limit.
We apply L’Hopital’s rule,
Since this limit exists, the application of L’Hopital’s was justified.