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Practice Quiz 1a

# Practice Quiz 1a - g(3 to make g continuous at x = 3...

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Practice Quiz IA for Calculus I, MATH 1501 (I) Let f ( x ) be given by f ( x ) = 1 3 + x 2 . (a) Find all x such that | f ( x ) - f (1) | < 1. (b) Find a δ ( ) > 0 such that | x - 1 | < δ ( ) ⇒ | f ( x ) - f (1) | < . and justify your answer. (II) Which of the following limits exist? If they don’t exist, explain why not. If they do, compute the value of the limit. (a) lim x 0 tan( 2 x ) x (b) lim x 1 ( x - 1) 2 ( x 2 - 1) 2 (c) lim x 1 x 3 - 1 x - 1 (d) lim x 1 sin( x ) - tan( x ) x 3 (III): Consider the following functions defined for x 6 = 3: f ( x ) = x - 3 x - 3 g ( x ) = x - 3 | x - 3 | . (a) What value, if any, can be assigned to f (3) to make f continuous at x = 3? Justify your answer. (b) What value, if any, can be assigned to
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Unformatted text preview: g (3) to make g continuous at x = 3? Justify your answer. (IV): Consider the sequence given recursively by a n +1 = √ a n + 1 and a 1 = 1 . (a) Explain why a n + 1 ≥ a 2 n for all n . (b) Is this sequence bounded? If so give a bound, and if not, explain why not. (Hint: Consider part (a) and the set of all x with x + 1 ≥ x 2 ). (c) Is this sequence convergent? If so, explain why and compute the limit. If not, explain why not....
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