Advanced Analysis by Min Yan

By Min Yan

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Since the fan OBP is sandwiched between the triangles OBP and OBQ, we have the inequality between the areas 1 1 π 1 sin x < x < tan x for 0 < x < . 2 2 2 2 Therefore we get 0 < sin x < x for 0 < x < and cos x < π , 2 sin x π < 1 for 0 < x < . 19) .... .. 1 ................................. .. .. ..... Q ..... P ............ . .. .................... . . . .... ...... .. ... ... .. . . ... ... . . . .. ... . . . ........ ......... .... .......... 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. From the two rules, further derive the following exponential rules. 1. l+∞ = 0 for 0 < l < 1. 3. (0+ )k = +∞ for k < 0.

Compare the infinities at +∞. √ x + 1 − 1, 3 √ x + 1 − 1, 1+ 1+ √ √ x∼ √ x at +∞. 1 + x, x2 (2 + cos x), xx , 2x . 31. Find α, so that the infinities at ∞ have the same order as xα . x(2 + cos x) + x2 (2 − sin x), 6x3 + 5x5 , x+ x2 + √ x3 . 32. Is it true that f1 (x) = o(g(x)) and f2 (x) = o(g(x)) =⇒ f1 (x) + f2 (x) = o(g(x))? Discuss the similar properties for product, composition, etc. Discuss the similar properties for other types of comparisons. 3. 33. Given a sequence of functions f1 (x), f2 (x), .

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