lecture13

# lecture13 - f c exists(iii lim x → c f x = f c A function...

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Math 006 (Lecture 13) Indeterminate Form Example 1. Find (a) lim x →- 2 x 2 - 4 x + 2 (b) lim x →- 1 x | x + 1 | x + 1 Definition 1. If lim x c f ( x ) = lim x c g ( x ) = 0, then lim x c f ( x ) g ( x ) is said to be indeterminate . Theorem 1. If lim x c f ( x ) = L = 0 and lim x c g ( x ) = 0 , then lim x c f ( x ) g ( x ) does not exist. Example 2. Evaluate each of the following limits: (a) lim x 1 x - 1 x 2 - 1 (b) lim x 1 ( x - 1) 2 x 2 - 1 (c) lim x 1 x 2 - 1 ( x - 1) 2 . 1

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Limit of a Difference Quotient A limit of the difference quotient is lim h 0 f ( a + h ) - f ( a ) h . Example 3. Find lim h 0 f ( a + h ) - f ( a ) h when (a) f ( x ) = 4 x - 5, a = 3 (b) f ( x ) = | x + 5 | , a = - 5 and a = 0 (c) f ( x ) = x , a = 2 2
Continuity Definition 2. A function f is continuous at the point x = c if (i) lim x c f ( x ) exists
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Unformatted text preview: f ( c ) exists (iii) lim x → c f ( x ) = f ( c ). A function is continuous on the interval a < x < b if it is continuous at each point on the interval. Example 4. Discuss the continuity of the function f at x =-1 , , 1 and 2. Example 5. Discuss the continuity of each function at the indicated point(s). (a) f ( x ) = x + 1 at x = 1 (b) g ( x ) = x 2-4 x-2 at x = 2 (c) h ( x ) = √ x at x =-1 (d) k ( x ) = | x | x at x = 1 (e) j ( x ) = x 1 3 at x =-1. 3...
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