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# Find the limit as x goes to 1 of log x / (x2 + x – 2)

Evaluate the limit.

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

Therefore,

# Find the limit as x goes to 0 of (ax – 1) / (bx – 1)

Evaluate the limit for .

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

Therefore, we have

# 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,

# Find the limit as x goes to 0 of (tan (2x)) / (sin (3x))

Evaluate the limit.

Using the previous exercise we know

Therefore, we can evaluate this limit as follows,

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

Evaluate the limit.

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

Therefore,

# 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,