By Min Yan

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**Additional resources for Advanced Analysis**

**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