This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 124 Quiz 2 Sept. 8, 2006 Name: 1. Show, using the rigorous definition of continuity, that the following piecewise function is not continuous at x = 0: g ( x ) = x  x  , x 6 = 0; , x = 0 Solution: Note that if x < 0 then x  x  = x x = 1 and if x > 0 then x  x  = x x = 1. Hence, lim x → g ( x ) = lim x → x  x  = lim x → ( 1) = 1 and lim x → + g ( x ) = lim x → + x  x  = lim x → + (1) = 1 This shows that lim x → g ( x ) does not exist, since the limit from the left does not equal the limit from the right. Now, if g were to be continuous at 0, it would be true that lim x → g ( x ) = g (0). However, since the limit on the left doesn’t even exist, this last equation has no hope of being true. We conclude that g is not continuous at 0. 2. Show mathematically that there is a number c with 0 ≤ c ≤ 1 such that f ( c ) = 0, where f ( x ) = 2 x 1 x . Solution: This is a case for the famous Intermediate Value Theorem. First note that f is continuous. Sinceis continuous....
View
Full Document
 Spring '08
 KENNEDY
 Calculus, Continuity, Continuous function, Limit of a function, lim g

Click to edit the document details