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Assignment 2 Solutions

# Assignment 2 Solutions - 1 F x lim x → 1 x 2 − 1 | x...

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MAT 1330, Fall 2010 Assignment 2 Due Wednesday October 6, 8:30am at the beginning of class. Late assignments will not be accepted; nor will unstapled assignments. Instructor (circle one): Frithjof Lutscher Angelika Welte Aziz Khanchi Robert Smith? DGD (circle one): 1 2 3 4 Student Name Student Number By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. Signature Question 1. Find lim x 0 x 3 + cos 5 x 10 , 000 . Answer: 1 10000 Give one sequence to support your claim. Five terms are enough. x n f ( x n ) 0.5 0.124920 0.1 0.001088 0.05 0.000222 0.01 0.000101 Question 2. Does the limit lim t 0 t 1 1 + t exist? Answer: Yes Justify your answer in at most two lines without using sequences of numerical values for x . Answer: lim t 0 t 1 1 + t = lim t 0 t 1 1 + t × 1 + 1 + t 1 + 1 + t = lim t 0 t (1 + 1 + t ) 1 (1 + t ) = lim t 0 t (1 + 1 + t ) t = lim t 0 (1 + 1 + t ) = 2 . 1

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Question 3. Let F ( x ) = x 2 1 | x 1 | . a) Find lim
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Unformatted text preview: 1 + F ( x ). lim x → 1 + x 2 − 1 | x − 1 | = lim x → 1 + x 2 − 1 x − 1 = lim x → 1 + ( x − 1)( x + 1) x − 1 = lim x → 1 + ( x + 1) = 2. b) Find lim x → 1-F ( x ). lim x → 1-x 2 − 1 | x − 1 | = lim x → 1-x 2 − 1 − ( x − 1) = lim x → 1-( x − 1)( x + 1) − ( x − 1) = lim x → 1-− ( x + 1) = − 2 . . c) Does lim x → 1 F ( x ) exist? Answer: No Justify your answer: lim x → 1 + F ( x ) ± = lim x → 1-F ( x ) = ⇒ The limit at 1 does not exist. Question 4. Consider the function f ( x ) = 3 x . Use the de±nition of the derivative to compute f ± (3). Answer: f ± (3) = lim h → f (3 + h ) − f (3) h = lim h → 3 3+ h − 1 h = lim h → 3 − (3+ h ) 3+ h h = lim h → − h h (3 + h ) = lim h → − 1 3 + h = − 1 3 . 2...
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