By Min Yan

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X 2. limx→0+ x(x ) . 3. limx→0+ (tan x)x . 4. limx→0 |x|tan x . 5. limx→0 (cos x)x . 1 7. limx→+∞ x x . 1 8. limx→+∞ (2x + 3x ) x . x 9. limx→+∞ (2x + 3x ) x2 +1 . 16. Prove limx→0 |p(x)|x = 1 for any nonzero polynomial p(x). 17. 27). 1 for sufficiently small x > 0. 18. 27). 19. Prove the following exponential rules. 1. l+∞ = +∞ for l > 1: If limx→a f (x) = l > 1 and limx→a g(x) = +∞, then limx→a f (x)g(x) = +∞. 2. (0+ )k = 0 for k > 0: If f (x) > 0, limx→a f (x) = 0 and limx→a g(x) = k > 0, then limx→a f (x)g(x) = 0.

18. 27). 19. Prove the following exponential rules. 1. l+∞ = +∞ for l > 1: If limx→a f (x) = l > 1 and limx→a g(x) = +∞, then limx→a f (x)g(x) = +∞. 2. (0+ )k = 0 for k > 0: If f (x) > 0, limx→a f (x) = 0 and limx→a g(x) = k > 0, then limx→a f (x)g(x) = 0. From the two rules, further derive the following exponential rules. 1. l+∞ = 0 for 0 < l < 1. 3. (0+ )k = +∞ for k < 0. 2. l−∞ = 0 for l > 1. 4. (+∞)k = 0 for k > 0. 3. 20. Provide counterexamples to the wrong exponential rules. (+∞)0 = 1, 1+∞ = 1, 00 = 1, 00 = 0.

If X = {−∞}, then sup X = −∞. Write down the similar statements for inf X. 37. For a not necessarily bounded sequence {xn }, extend the definition of LIM{xn } by adding +∞ if there is a subsequence diverging to +∞, and adding −∞ if there is a subsequence diverging to −∞. 36. 11. 1. A sequence with no upper bound must have a subsequence diverging to +∞. This means limn→∞ xn = +∞. 2. If there is no subsequence with finite limit and no subsequence diverging to −∞, then the whole sequence diverges to +∞.