Evaluate the limit.

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

Therefore,

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Stumbling Robot

A Fraction of a Dot
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Tag: o-notation

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Evaluate the limit for .

First we write and . Then we use the expansion of (p. 287), to obtain expansions for and ,

Therefore, we have

Evaluate the limit.

We know (p. 287) the following expansions as ,

(Note that these are the same expansion when we use only the first order terms. This tells us that and behave similarly near 0. We would need to take higher order terms to differentiate between the two. For instance, if we wanted to include cubic terms we would have , but .) From here we compute the limit,

Evaluate the limit.

To evaluate this we use the trig identity to simplify

We have proved the limit earlier (at least once), but let’s do it again using the techniques of this section and -notation.

Evaluate the limit.

Using the expansions as (p. 287)

we compute,

Evaluate the limit.

Using the expansion (p. 287)

we compute,

Evaluate the limit.

Using the previous exercise we know

Therefore, we can evaluate this limit as follows,

Evaluate the limit.

We know (p. 287) that the expansion for is given by

Therefore,

Using the expansion

prove that

Using a similar method, find

*Proof.* Since

we have

(We use Theorem 7.8(c) since in the second line to bring the inside the little .)

For the second limit, we use the Taylor expansion for and replace with to obtain,

Find constants , and such that

From the Taylor formula for we have

Replacing by we then have

Therefore,