Arithmetic with large and small numbers

Do not use a calculator for any of these problems. Write your answers on this or a separate sheet. Give brief explanations.

• 1. If 0 < x < 1, then is 5/x less than or greater than 5 ?

  2. If 0 < x < 1, then is 0.01/x less than or greater than 0.01?

  3. If 0 < x < 1, then is x^9 less than or greater than x?

  4. If 0 < x < 1, then is x(1.01) less than or greater than x?

  5. Is (1.01)^2 less than or greater than 1.01 ?

  6. As x --> infinity, (1.01)^x --> ?

  7. Is (0.99)^2 less than or greater than 0.99 ?

  8. As x --> infinity, ( 0.99)^x --> ?

  9. Is the square root of 5 less than or greater than 1?

  10. How about the hundredth root of 5? Why?

  11. How about 5^(0.000001)? Why?

  12. As x --> infinity, x^(0.00001) --> ?

  13. As x --> infinity, 100000^(1/x) --> ?

  14. True or False: if L is large and S is close to 0, then LS is necessarily close to 0.

  15. True or False: if L is large, and S is close to 0, then L^S is necessarily close to 1. (Compare #10 & #11.)

  16. True or False: if b and c are both close to 1, then b^c is necessarily close to 1.

  17. Fill in the blank: if b and c are both positive and close to 0, then b^c is necessarily close to _____.

  18. Fill in the blank: if b is close to 1 and c is close to 0, then b^c is necessarily close to _____.

  19. Fill in the blank: if b is close to 1, and if c is positive and close to 0, then c^b is necessarily close to _____.

  20. Fill in the blank: if b is close to 1 and L is large, then b^L is necessarily close to _____.

  21. Fill in the blank: if b is close to 1 and L is large, then L^b is necessarily close to _____.

  22. Order the following expressions in increasing order, assuming x=9^(9^99):

  23. x^2; 2^x; e^x; x^e; ln(x); 1/x; e^(-x); sqrt(x); x^0.01, e^(-x^2).

  24. Hint: When x is very large, e^x > x^constant > ln(x) (assuming the constant is positive).