### SOLUTIONS TO LIMITS USING L'HOPITAL'S RULE

SOLUTION 1 :

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(Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately.)

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truein truein SOLUTION 3: = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately.) truein

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truein truein SOLUTION 4: = = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately. Recall that .) truein

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SOLUTION 6: = = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate the top using the product rule and the bottom using the chain rule.) truein

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truein truein SOLUTION 7: = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate the top using the chain rule and the bottom using the chain rule. Recall that and .) truein

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truein truein SOLUTION 8: = = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate the top using the chain rule and the bottom using the product rule.) truein

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truein truein SOLUTION 9: = = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate the top using the product rule and the bottom using the chain rule.) truein

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truein truein SOLUTION 10: =

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truein (Applying Theorem 1 for l'Hopital's Rule again leads to . Stop and think. This is getting nowhere. Go back to the beginning and algebraically rewrite the problem by flipping" both numerator and denominator.)

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truein truein SOLUTION 11: = = =

truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top using the chain rule and the bottom.) truein

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truein truein SOLUTION 12: = = =

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truein truein SOLUTION 13: =

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truein truein SOLUTION 14: = = =

truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom using the chain rule.) truein

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truein truein SOLUTION 15: = = =

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truein (Applying Theorem 2 for l'Hopital's Rule again will lead to . Stop and think. This is going nowhere. There is an easy way out.)

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truein truein SOLUTION 16: = = =

truein (Apply Theorem 2. Differentiate the top and the bottom.) truein

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truein (Applying Theorem 2 for l'Hopital's Rule again will get us nowhere. Stop and think. Go back to the original problem and rewrite it.) truein

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truein truein SOLUTION 17: =

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truein truein SOLUTION 18: = =

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truein truein SOLUTION 19: = = =

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truein truein SOLUTION 20: = =

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truein truein SOLUTION 21: = =

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truein truein SOLUTION 22: = =

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truein truein SOLUTION 23: = =

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truein truein SOLUTION 24: = =

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truein truein SOLUTION 25: = =

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truein truein SOLUTION 26: = =

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truein truein SOLUTION 28: b.) =

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truein truein SOLUTION 28: c.) The answers to parts a.) and b.) tell us that l'Hopital's Rule may give us a wrong answer if the answer is  does not exist." We can only be sure that l'Hopital's Rule gives us the correct answer if the answer is finite, , or .

Duane Kouba 2008-12-09