This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 18.014 Problem Set 4 Solutions Total: 24 points Problem 1: Establish the following limit formulas. You may assume the formula lim x sin( x x ) = 1. (a) sin(5 x ) lim = 5 . x sin( x ) (b) sin(5 x ) sin(3 x ) lim = 2 . x x (c) 2 1 1 1 x lim = . x x 2 2 Solution (4 points) (a) Using the product formula for limits (Thm. 3.1 part iii), we have sin(5 x ) sin(5 x ) 5 x sin(5 x ) 5 x lim = lim = lim lim = AB. x sin( x ) x 5 x sin( x ) x 5 x x sin( x ) For the first term, note that 5 x approaches zero as x approaches zero; hence, 5 x A = 1 by the assumed limit formula. For the second term, note lim x sin( x ) = 5 lim x sin( 1 x ) /x = 5 1 = 5 by the product rule and the quotient rule (Thm. 3.1 part iv). Thus, B = 5 and sin(5 x ) lim = 5 x sin( x ) as desired. (b) Here we use the difference rule (Thm. 3.1 part ii) to obtain sin(5 x ) sin(3 x ) sin(5 x ) sin(3 x ) lim = lim lim . x x x x x x Next, for any real number a = 0, we observe sin( ax ) sin( ax ) sin( x ) lim = a lim = a lim = a. x x x ax x x 1 Plugging back into the above formula yields sin(5 x ) sin(3 x ) sin(5 x ) sin(3 x ) lim = lim lim = 5 3 = 2 . x x x x x x (c) We use the product rule to get 2 1 1 x (1 1 x 2 )(1 + 1 x 2 ) 1 lim = lim lim . x x 2 x x 2 x (1 + 1 x 2 ) Since (1 1 x 2 )(1 + 1 x 2 ) = 1 (1 x 2 ) = x 2 , the first limit is one. For the 1 2 second limit, note that is the composition of the functions 1 x , x , 1+ x , 2 1+ 1 x and x 1 , which are continuous at the points 0, 1, 1, and 2 by Example 5 and Theorem 3 . 2. Hence, by Theorem 3 . 5, the function 1+ 1 1 x 2 is continuous at x = 0, and we can just plug in x = to get that the limit of the second term is 1 / 2. Multiplying...
View
Full
Document
This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.
 Fall '10
 ChristineBreiner
 Calculus, Formulas

Click to edit the document details