= =
(Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately.)
=
truein
=
truein
= . truein truein SOLUTION 2: = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately.) truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 3: = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately.) truein
=
truein
=
truein
= .
truein truein SOLUTION 4: = = = truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate top and bottom separately. Recall that .) truein
=
truein
=
truein
=
truein
=
truein (Recall that .) truein truein
= . truein truein SOLUTION 5: = = truein (Apply THEOREM 1 for l'Hopital's Rule. Differentiate top and bottom separately.) truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein
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
=
truein
=
truein
=
truein
=
truein
= .
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
=
truein
=
truein
=
truein
= .
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
=
truein
= = =
truein (Apply Theorem 1 for l'Hopital's Rule again. Differentiate the top using the product rule and the chain rule and the bottom using the product rule and the chain rule.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
= .
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
=
truein
= = =
truein (Apply Theorem 1 for l'Hopital's Rule again. Differentiate the top using the product rule and the bottom using the product rule and the chain rule.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 10: =
truein
= = = =
truein (Apply Theorem 1 for l'Hopital's Rule. Differentiate the top using the chain rule and the bottom. Recall that .) truein
=
truein
=
truein
= = .
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.)
truein =
truein = = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
=
truein =
truein = .
truein truein SOLUTION 11: = = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top using the chain rule and the bottom.) truein
=
truein
=
truein
=
truein
= (So limit does not exist.).
truein truein SOLUTION 12: = = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom.) truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 13: =
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom.) truein
=
truein
= =
truein (Apply Theorem 2 for l'Hopital's Rule again.) truein
truein
=
truein
= .
truein truein SOLUTION 14: = = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom using the chain rule.) truein
=
truein
=
truein
= = = =
truein (Apply Theorem 2 for l'Hopital's Rule again.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
= truein
= .
truein truein SOLUTION 15: = = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom.) truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule again.)
truein
=
truein
=
truein
= = =
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.)
truein
=
truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 16: = = =
truein (Apply Theorem 2. Differentiate the top and the bottom.) truein
= = =
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
=
truein
= =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and bottom using the product rule.)
truein
=
truein
=
truein
= .
truein truein SOLUTION 17: =
truein (Rewrite the expression by using a conjugate to circumvent this indeterminate form.) truein
=
truein (Recall that .) truein
=
truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule. Use the chain rule in the bottom.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
= (The limit does not exist.) .
truein truein SOLUTION 18: = =
truein (This is an indeterminate form. ``Flip" so that l'Hopital's Rule can be applied.) truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule. Differentiate the top and the bottom separately.) truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 19: = = =
truein (This is an indeterminate form. ``Flip" so that l'Hopital's Rule can be applied.) truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
= =
truein (Apply Theorem 2 for l'Hopital's Rule again.) truein
=
truein
=
truein
= .
truein truein SOLUTION 20: = =
truein (Rewrite the expression to circumvent this indeterminate form. Recall that .) truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein (Recall that .) truein
=
truein
=
truein
= = = =
truein (Apply Theorem 1 for l'Hopital's Rule.)
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 21: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
= = =
truein (``Flip" to circumvent this indeterminate form.) truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
=
truein
= = = =
truein (Apply Theorem 1 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 22: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
= = =
truein (``Flip" to circumvent this indeterminate form.) truein
= = =
truein (Apply Theorem 1 for l'Hopital's Rule. Recall that ) truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 23: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
= = =
truein (Apply Theorem 1 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 24: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
=
truein
= = =
truein (``Flip" to circumvent this indeterminate form.) truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
= = =
truein (Apply Theorem 1 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 25: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 26: = =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
=
truein
= = =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 27: =
truein (Rewrite the problem to circumvent this indeterminate form. Recall that ) truein
=
truein
=
truein
=
truein
=
truein (See Example 4 in the material preceeding this problem set.) truein
= truein
=
truein
=
truein
=
truein
= .
truein truein SOLUTION 28: a.) Start with , truein so that
truein
truein and (Assume that .)
truein .
truein Then =
truein
=
truein
=
truein
=
truein
= .
truein Similarly, = .
truein Since
truein = = ,
truein it follows from the Squeeze Principle that
truein = .
truein truein SOLUTION 28: b.) =
truein (Apply Theorem 2 for l'Hopital's Rule.) truein
=
truein which does not exist since truein
.
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 .