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Limit Laws

# Limit Laws - CALCULATING LIMITS USING THE LIMIT LAWS Limit...

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CALCULATING LIMITS USING THE LIMIT LAWS Limit Laws Suppose that c is a constant and the limits lim x a f ( x ) and lim x a g ( x ) exist. Then, 1 . lim x a [ f ( x ) + g ( x )] = lim x a f ( x ) + lim x a g ( x ) 2 . lim x a [ f ( x ) - g ( x )] = lim x a f ( x ) - lim x a g ( x ) 3 . lim x a [ cf ( x )] = c lim x a f ( x ) 4 . lim x a [ f ( x ) g ( x )] = lim x a f ( x ) · lim x a g ( x ) 5 . lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) if lim x a g ( x ) 6 = 0 6 . lim x a [ f ( x )] c = h lim x a f ( x ) i c If f is a polynomial or a rational function and a is in the domain of f , then lim x a f ( x ) = f ( a ) . We call this Direct Substitution Property , and functions with the Direct Substitution Property are called continuous at a . By using this property and the limit laws, we can compute most of limit problems. Example 1. Find lim h 1 h 3 - 1 h - 1 . Solution We cannot use Direct Substitution Property because the function is not defined at h = 1. But we can simplify the function algebraically and get h 3 - 1 h - 1 = h 2 + h + 1 when h 6 = 1. Thus, lim h 1 h 3 - 1

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