Math 31B WorksheetWeek 3

1. Suppose a is a positive constant. Compute the limit

limxæŒ

ln(x)xa

What does this limit tell you about the growth of these functions? (You should answerusing π or ∫, and describing your answer in plain English.)

2. (a) Repeat the previous problem, but this time comparing the functions (ln(x))2 andxa. (Hint: You’ll have to do L’Hôpital’s Rule twice this time.)

(b) Repeat the previous problem again, but this time comparing the functions (ln(x))3and xa. (Hint: You’ll have to do L’Hôpital’s Rule three times.)

(c) Based on the work you did for the previous two parts, do you see a pattern?What would you conclude about the growth of (ln(x))N (for any positive integerN) compared to xa (for any positive constant a)?

(a) Compute limnæŒ

✓4 ≠ 3

n4

◆.

(b) Use your answer to part (a) to compute limnæŒ

✓4 ≠ 3

n4

◆ 12.

What limit law are you using here? (You may want to discuss with your TA/LA,and/or look through the theorems in section 11.1 in your textbook.)

3. Consider the sequence defined by the recurrence relation

a1 = 1, an = 2an≠1 ≠3

an≠1

Compute a2, a3, a4, and a5. (Feel free to use a calculator.)

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thx a a

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