Calculus II

Calculus Practice Exams

Answers.

Problem 1. Find the limit

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Answer. Note that we have an indeterminate form . You may want to try Hopital rule and see how difficult the calculations become except if you use the substitution . Here we will use Taylor Polynomials to find this limit. We have

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This gives

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Clearly the limit is

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Problem 2. Find the limit

Answer. Note that we have an indeterminate form . Since we a have a square root in the function, we will use the conjugate forms to get rid of it. We have

Since

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(note that |x| = -x because and therefore x < 0), hence

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It is now obvious that

Problem 3. Find

Answer. Note that we have an indeterminate form . In order to find the limit, we will use the formula

We can assume that . In this case, we have

Since

we have to concentrate on the term . We have

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Since

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we get

Therefore we have

Problem 4. Determine the convergence or divergence of

(a)

Answer. The bad points are , , -1.
At , we have

when . Recall that

when if and only if

Using the limit test and the p-test, we know that

are both convergent. Thereofre

are both convergent.
At -1, we have

when . Again using the limit test and the p-test, we know that

is divergent. Therefore the improper integral

is divergent.

(b)

Answer. The bad points are 0 and .
At , we have

when . Using the limit test and the p-test, we know that

is divergent. Therefore

is divergent. So the improper integral

is divergent. We suggest to consider the other bad points too though there is no need for that. But you never know, your previous conclusion may be wrong!!!!
At 0, we have