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# Show that the limit and derivative cannot be interchanged when fn(x) = (sin (nx)) / n

For each positive integer and and all real define

Prove that

Proof. First, we have

for all . Hence,

On the other hand,

Hence,

# Prove some properties of the function e-1/x2

Consider the function

and .

1. Prove that for every positive number we have

2. Prove that if then

where is a polynomial in .

3. Prove that

1. Proof. (A specific case of this general theorem is actually the first problem of this section, here. Maybe it’s worth taking a look since this proof is just generalizing that particular case.) We make the substitution , so that as and we have

by Theorem 7.11 (page 301 of Apostol) since implies as well

2. Proof. The proof is by induction on . In the case we have

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 .

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

3. Proof. The proof is by induction on . If then we use the limit definition of the derivative to compute the derivative at 0,

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 limit of the given function

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 limit of the given function

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 limit of the given function

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 limit of the given function

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 limit of the given function

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 limit of the given function

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 limit of the given function

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 limit of the given function

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.