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# Find the limit as x goes to 0 of (sin x) / (arctan x)

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,

# Find the limit as x goes to 0 of (1 – cos2 x) / (x tan x)

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.

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

Evaluate the limit.

Using the expansions as (p. 287)

we compute,

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

Evaluate the limit.

Using the expansion (p. 287)

we compute,

# Prove some limits of cos x

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 an expression for log x as a quadratic polynomial in (x-1)

Find constants , and such that

From the Taylor formula for we have

Replacing by we then have

Therefore,

# Find a polynomial of minimal degree such that sin (x – x2) = P(x) + o(x6)

Find the polynomial of minimal degree such that

Using the Taylor expansion of we know as we have

(This is where we see how nice -notation can be. All of the terms in the polynomials larger than will get absorbed into the . This simplifies computations tremendously when we don’t care about the higher order terms.) Therefore, we have

# Find a cubic polynomial such that x cos x = P(x) + o((x-1)3)

Find a cubic polynomial such that

Let and take some derivatives,

Therefore, the Taylor polynomial approximation to around 1 is given by

Therefore, we can take

# Find a quadratic polynomial such that 2x = P(x) + o(x2)

Find a quadratic polynomial such that

Since and

we have

Therefore we may take

# Compute π using the Taylor polynomial of arctan x

For this exercise define

1. Using the trig identity

twice, once with , and then the second time with , show that

Then use the same identity again with and to show

This establishes the identity

2. Using the Taylor polynomial approximation at prove that

3. Using the Taylor polynomial approximation at prove that

4. Using the above parts show that the value of to seven decimal places is 3.1415926.

1. Proof. Letting we have

Letting we have

Letting and we have (recalling that )

But then

2. Proof. We know the Taylor polynomial approximation for from this exercise (Section 7.8, Exercise #3):

Therefore, we can compute an approximation to ,

where

Therefore,

3. Proof. Again using the Taylor polynomial approximation to we have

4. Finally,