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sa118B_1

# sa118B_1 - < x< 1 such that f x = g x 5 Evaluate(a lim...

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Self-Assessment Questions – Math 118B, Winter 2010 1. Let D R n be a bounded set and let f be uniformly continuous on D . Prove that f is bounded on D . 2. Let f be a function deFned on a set E which is such that it cen be uni- formly approximated within ǫ on E by functions F that are uniformly continuous on E , for every ǫ > 0. Show that f must itself be uniformly continuous on E . 3. Prove that if f is continuous on a compact set D , then f ( D ) is also compact. 4. Let f and g be continuous on [0 , 1] and suppose that f (0) < g (0) and f (1) > g (1). Prove that there is a point 0

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Unformatted text preview: < x < 1 such that f ( x ) = g ( x ). 5. Evaluate: (a) lim x → 1-cos( x 2 ) x 3 sin x . (b) lim x → 0+ x x . (c) lim x → sin x + cos x-e x log(1 + x 2 ) . 6. Show that n s i =1 i = n ( n + 1) 2 , and use it to prove that i 1 x dx = 1 2 . 1 2 7. Show that n s i =1 i 2 = n ( n + 1)(2 n + 1) 6 , and use it to prove that i 1 x 2 dx = 1 3 . 8. Prove that lim n →∞ 1 n k +1 n s i =1 i k = 1 k + 1 , for al l k ∈ N ....
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sa118B_1 - < x< 1 such that f x = g x 5 Evaluate(a lim...

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