PracticeMidtermII

PracticeMidtermII - MAT 319/320 Practice Midterm II 1...

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Unformatted text preview: MAT 319/320 Practice Midterm II 1 Problem 1. Define a sequence (xn) by : x0 = 1 and xn +1 = 3 xn + 1. Does (xn) have a limit? If yes, what is this limit? Problem 2. Is the infinite series the limit). 1 +∞ √ n =1 n + n convergent? (If yes, you do not need to find the value of Problem 3. Recall the definition of the continuity of a function f at a point c. Problem 4. What is limx→ + ∞ √ x+3 ? x+7+1 Problem 5. Use the definition of a limit (I mean use “ε, δ ”) to prove that 5x2 + 2x + 1 = 2. How could you prove the same thing using an easier way? limx→− 1 x+3 Problem 6. Let f : [0, 3] → R be a continuous function. Assume that f (1) > 0, then prove the existence of a small δ -neighborhood of 1 on which the function f has no root (meaning there is no x in this neighborhood such that f (x) = 0). 1 ...
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