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# Find the limit as x goes to 0 of (x + e2x)1/x

Evaluate the limit.

From the definition of the exponential we have

So, first we use the expansion of as (page 287 of Apostol) to write

Therefore, as we have

Now, since as we can use the expansion (again, page 287) of as to write

Therefore, as we have

So, getting back to the expression we started with,

But, as in the previous exercise (Section 7.11, Exercise #23) we know . Hence,

# Find the limit as x goes to 1 of x1 / (1-x)

Evaluate the limit.

First, we write

From this exercise (Section 7.11, Exercise #4) we know that as we have

Therefore, as ,

So, we then have

(Here we could say that since the exponential is a continuous function we can bring the limit inside and so this becomes . I’m not sure we know we can pass limits through continuous functions like that, so we continue on with expanding the exponential as in previous exercises.)
Since as we take the expansion of as ,

Therefore,

# Find the limit as x goes to 0 of (cos (sin x) – cos x) / x4

Evaluate the limit.

We know (page 287 of Apostol) that the expansions for and as are given by

Therefore, we have the following expansion for as ,

So, now we can take the limit,

# Find the limit as x goes to 0 of (ax – asin x) / x3

Evaluate the limit.

First, we want to get expansions for and as . For we write and use the expansion (page 287 of Apostol) of . This gives us

Next, for , again we write and then use the expansion for we have

Now, we need use the expansion for (again, page 287 of Apostol)

and substitute this into our expansion of ,

(Again, this is the really nice part of little -notation. We had lots of terms in powers of greater than 3, but they all get absorbed into , so we don’t actually have to multiply out and figure out what they all were. We only need to figure out the terms for the powers of up to 3. Of course, the 3 could be any number depending on the situation; we chose 3 in this case because we know that’s what we will want in the limit we are trying to evaluate.)

So, now we have expansions for and (in which most of the terms cancel when we subtract) and we can evaluate the limit.

# Evaluate the limit as x goes to 0 of a given function

Evaluate the limit.

We use the expansions for (given in Example 1 on page 288 of Apostol) and (on page 287 of Apostol):

Therefore, we have

# Find the limit as x goes to 0 of (cosh x – cos x) / x2

Evaluate the limit.

We use the definition of in terms of the exponential:

and the expansions (page 287 of Apostol) of and as :

Putting these together we evaluate the limit:

# Find the limit as x goes to 1 of given function

Evaluate the limit.

Using the expansion (from this exercise, Section 7.11, Exercise #4)

we compute the limit as follows:

# Find the limit as x goes to (π / 2) of cos x / (x – (π / 2))

Evaluate the limit.

Since we use the expansion (page 287) for ,

Therefore,

# Find the limit as x goes to 0 of the given expression

Evaluate the limit.

We use the expansions on page 287 for and ,

Then we compute the limit,

# Evaluate the limit as x goes to 0 of (1 – cos (x2)) / (x2 sin (x2))

Evaluate the limit.

We use the expansions (p. 287) for and as to write,

Therefore, we compute the limit as