3 Trigonometric functions and in verse trigonometric functions 4 Exponential

3 trigonometric functions and in verse trigonometric

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3. Trigonometric functions and in- verse trigonometric functions.
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4. Exponential and logarithmic func- tions. 5. Combinations of the above.
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Section 5.1: Limits of Functions Def. Let f : D R and let c be an accumulation point of D . We say that lim x c f ( x ) = L if for every > 0 there exists δ > 0 such that | f ( x ) - L | < whenever x D and 0 < | x - c | < δ.
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In other words, lim x c f ( x ) = L means . . . For every > 0, there exists δ > 0 such that if x N * ( c, δ ) D , then f ( x ) N ( L, ).
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Equivalently: Def. Let f : D R and let c be an accumulation point of D . A number L is the limit of f at c if for every neighborhood V of L there exists a deleted neighborhood U of c such that f ( U D ) V . Notation lim x c f ( x ) = L .
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THEOREM 1: Let f : D R and let c be an accumulation point of D . If lim x c f ( x ) = L exists, then it is unique. That is, f can have only one limit at c .
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Examples: 1. lim x 3 (5 x - 3) = 12. 2. lim x 2 2 x 2 + 4 x - 16 x - 2 = 12. 3. lim x 5 ( x 2 - 3 x + 1) = 11.
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THEOREM 2: Let f : D R and let c be an accumulation point of D . Then lim x c f ( x ) = L if and only if f ( s n ) L for every sequence ( s n ) in D such that s n c and s n 6 = c for all n N .
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THEOREM 3: Let f : D R and let c be an accumulation point of D . The following are equivalent: 1. lim x c f ( x ) does not exist. 2. There exists a sequence ( s n ) in D such that s n c , but ( f ( s n )) does not converge.
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THEOREM 4: If lim x c f ( x ) = L, then there exists δ > 0 such that f is bounded on N * ( c, δ ). That is, there is a number M such that | f ( x ) | ≤ M for all x N * ( c, δ ) D.
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THEOREM 5: (Arithmetic) Let f, g : D R and let c be an accumulation point of D . If lim x c f ( x ) = L and lim x c g ( x ) = M, then 1. lim x c [ f ( x ) + g ( x )] = L + M , 2. lim x c [ f ( x ) - g ( x )] = L - M , 3. lim x c [ f ( x ) g ( x )] = LM ,
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4 lim x c [ k f ( x )] = kL, k constant, 5 lim x c f ( x ) g ( x ) = L M provided M 6 = 0.
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  • Topology, Mathematical analysis, Metric space, Limit of a sequence

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