Explanation Given: • lim x → 1 x 1 x − 1 Key Concepts Used: Limit: Assume that f ( x ) is defined for all x in an open interval containing c , but not necessarily at c itself. We say that the limit of f ( x ) as x approaches c is equal to L . If | f ( x ) − L | becomes arbitrarily small when x is any number sufficiently close (but not equal) to c . In this case, we write lim x → c f ( x ) = L . If the value of f ( x ) do not converge to any limit as x → c , we say that lim x → c f ( x ) does not exist. The limit may exist even if f ( c ) is not defined. Indeterminate form: We say that f ( x ) has an indeterminate form at x = c if the formula for f ( c ) yields an undefined expression of the type 0 0 , ∞ ∞ , ∞ ⋅ 0 , ∞ − ∞ . When this kind of indeterminate form occurs, then we have to transform f ( x ) algebraically, if possible, into a new expression that is defined and continuous at x = c and then evaluate the limit by substitution. It is possible by cancelling the common factor from the numerator and denominator at the appropriate moment, thereby removing the indeterminacy. Otherwise we can use L’Hopital’s Rule to remove this indeterminacy. L’Hopital’s Rule: Assume that f ( x ) and g ( x ) are differentiable on an open interval containing a and that f ( a ) = g ( a ) = 0 . Also assume that g ' ( x ) ≠ 0 (except possible at a ). Then lim x → a f ( x ) g ( x ) = lim x → a f ' ( x ) g ' ( x ) if the limit on the rights exists or is infinite ( ∞ o r − ∞ ). Determining the Limit numerically: x 0.75 0.90 0.99 0.999 1 1.001 1.01 1.11 1.25 x 1 x − 1 3.1604 2.8679 2.7319 2.7196 ? 2.7169 2.7048 2.5824 2.4414 Formula Used: L’Hopital’s Rule: lim x → a f ( x ) g ( x ) = lim x → a f ' ( x ) g ' ( x ) Calculation: Description: lim x → 0 1 − cos 3 ( x ) x 2 Step 1 : Write the given problem of limit. lim x → 1 x 1 x − 1 Step 2 : Substitute the value of limit for the problem. lim x → 1 x 1 x − 1 = 1 1 1 − 1 = 1 1 0 = 1 ∞ Step 3 :As we obtain an indeterminate form of type 1 ∞ , we transform f ( x ) algebraically, if possible, into a new expression that is defined and continuous at x = c and then evaluate the limit by substitution

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 2, Problem 6CRE

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The limit numerically to two decimal places if not statement that the limit does not exist

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Chapter 2, Problem 5CRE

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Chapter 2 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E

Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.2 - Prob. 1PQ Ch. 2.2 - Prob. 2PQ Ch. 2.2 - Prob. 3PQ Ch. 2.2 - Prob. 4PQ Ch. 2.2 - Prob. 5PQ Ch. 2.2 - Prob. 6PQ Ch. 2.2 - Prob. 7PQ Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Prob. 16E Ch. 2.2 - Prob. 17E Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - 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Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - Prob. 35E Ch. 2.6 - Prob. 36E Ch. 2.6 - Prob. 37E Ch. 2.6 - Prob. 38E Ch. 2.6 - Prob. 39E Ch. 2.6 - Prob. 40E Ch. 2.6 - Prob. 41E Ch. 2.6 - Prob. 42E Ch. 2.6 - Prob. 43E Ch. 2.6 - Prob. 44E Ch. 2.6 - Prob. 45E Ch. 2.6 - Prob. 46E Ch. 2.6 - Prob. 47E Ch. 2.6 - Prob. 48E Ch. 2.6 - Prob. 49E Ch. 2.6 - Prob. 50E Ch. 2.6 - Prob. 51E Ch. 2.6 - Prob. 52E Ch. 2.6 - Prob. 53E Ch. 2.6 - Prob. 54E Ch. 2.6 - Prob. 55E Ch. 2.6 - Prob. 56E Ch. 2.6 - Prob. 57E Ch. 2.6 - Prob. 58E Ch. 2.6 - Prob. 59E Ch. 2.6 - Prob. 60E Ch. 2.6 - Prob. 61E Ch. 2.7 - Prob. 1PQ Ch. 2.7 - Prob. 2PQ Ch. 2.7 - Prob. 3PQ Ch. 2.7 - Prob. 4PQ Ch. 2.7 - Prob. 5PQ Ch. 2.7 - Prob. 6PQ Ch. 2.7 - Prob. 1E Ch. 2.7 - Prob. 2E Ch. 2.7 - Prob. 3E Ch. 2.7 - Prob. 4E Ch. 2.7 - Prob. 5E Ch. 2.7 - Prob. 6E Ch. 2.7 - Prob. 7E Ch. 2.7 - Prob. 8E Ch. 2.7 - Prob. 9E Ch. 2.7 - Prob. 10E Ch. 2.7 - Prob. 11E Ch. 2.7 - Prob. 12E Ch. 2.7 - Prob. 13E Ch. 2.7 - Prob. 14E Ch. 2.7 - Prob. 15E Ch. 2.7 - 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The limit numerically to two decimal places if not statement that the limit does not exist

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Chapter 2 Solutions