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Advanced Analysis by Min Yan

By Min Yan

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Example text

Where the inequality 1. limx→∞ 2. limx→∞ x2 x = 0. +1 sin x = 0. x 3. limx→∞ x2 − 1 = 1. x2 + 1 4. limx→∞ x2 + x − 1 = 1. 8. Prove lim αx = 0 for α > 1. 9. 8). The final variation is the divergence to infinity. 5. A function f (x) diverges to infinity at a, denoted limx→a f (x) = ∞, if for any b, there is δ > 0, such that 0 < |x − a| < δ =⇒ |f (x)| > b. 11) The divergence to positive and negative infinities, denoted limx→a f (x) = +∞ and limx→a f (x) = −∞ respectively, can be similarly defined.

X1 x2 x3 x4x5 b .. 16. Prove the inductively defined sequences converge and find the limits. 1. x1 = 1, xn+1 = 1 + 2 . xn 2. x1 = 1, xn+1 = x2n + 2 . 2. 17. Discuss the convergence of the sequence defined by x1 = α, xn+1 = γ β + γxn . Do the same to the sequence defined by x1 = α, xn+1 = β + . 18. Let a, b > 0. Define sequences by a1 = a, b1 = b, an = an−1 + bn−1 a+b 2an−1 bn−1 2ab .

X→1− x→1 Since the left and right limits are not equal, g(x) diverges at 1. 44 CHAPTER 1. 6. If α > 0, then for any 0 0, we have =⇒ 0 < xα < δ α = . This shows limx→0+ xα = 0. 5. Let [x] be the biggest integer n satisfying n ≤ x. 2] = −4. Compute the limits. 1. limx→2+ [x]. 3. limx→√2+ [x]. 2. limx→2− [x]. 4. limx→√2− [x]. √ 5. limx→4+ [ x]. √ 6. limx→4− [ x]. 6. Determine whether limits exist. 1. limx→0 √ x √ − −x 2. limx→1 2 if x > 1 . x if x < 1 if x > 0 . if x < 0 3. limx→1 1 x if x > 1 .

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