PracticeFinal319

PracticeFinal319 - > such that f is bounded on [...

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MAT 319 Practice FINAL Problem 1. What is the limit of ( x n ) = n 3 n ! ? Problem 2. Use the deFnition of the limit to prove that lim n 2 1 3 n 2 + 1 = 1 3 . Problem 3. Prove that an increasing sequence that is bounded above is necessarily converging. Problem 4. Show that if u n is unbounded then there is a subsequence u n k of terms that are all non zero and such that 1 u n k 0 . Problem 5. Is the inFnite series n =1 1 n 2 n + 2 convergent? Problem 6. Evaluate the following limit, or show that it doesn’t exist: lim x + x x 2 x + x. x . Problem 7. Assume that f : R R is such that: for any x R there is a
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Unformatted text preview: > such that f is bounded on [ x-, x + ] . Is the function f bounded on R ? (If yes, prove it; if not give a counter-example). Problem 8. Is the function g : R R deFned by g ( x ) = 3 x + | x | dierentiable everywhere? (Prove your assertion!) Problem 9. If f : R R is dierentiable at c R , then show that lim ( n ( f ( c + 1 n )-f ( c )) exists and is equal to f ( c ) . Problem 10. Show that if x > then we have 1 + x 3 l 1 + 1 3 x 1...
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