ConcepTest Section 4.7 Question 1 True or false?

ConcepTest Section 4.7 Answer 1 ANSWER COMMENT: Follow-up Question. What does l’Hopital’s rule say? Answer. If f and g are differentiable, f (a) = g(a) = 0, and g’(a) ≠ 0, then (a) False. l’Hopital’s rule concerns the equality of limits, not the equality of functions. (b) False. In applying l’Hopital’s rule, the numerator and the denominator are differentiated separately. (c) False. To apply l’Hopital’s rule to a product, the expression must first be rewritten as a ratio.

Consider 3 limits: Which is the correct ranking? (a)p < q < r (b)p < r < q (c)q < p < r (d)q < r < p (e)r < p < q (f)r < q < p ConcepTest Section 4.7 Question 2

ConcepTest Section 4.7 Answer 2 ANSWER (a) Since p is a limit of the form 0/0 it can be evaluated by l’Hopital’s rule. The limit for q is the limit of a function that is continuous at x = 1, so l’Hopital’s rule does not apply. To compute the limit, evaluate the function.

Since r is a limit of the form 0/0 it can be evaluated by l’Hopital’s rule. We have COMMENT: Students may attempt to evaluate q with l’Hopital’s rule. Remind them that l’Hopital’s rule usually gives the wrong answer when it does not apply. ConcepTest Section 4.7 Answer 2 ANSWER (cont’d)

ConcepTest Section 4.7 Question 3 For which of the following can you use l’Hopital’s rule to evaluate the limit?

ConcepTest Section 4.7 Answer 3 ANSWER COMMENT: Follow-up Question. Compute the limits. Answer. For (a) the limit does not exist; for (b) the limit is 1; for (c) and (d) the limit is zero.

Consider 3 limits: Which is the correct ranking? (a)p < q < r (b)p < r < q (c)q < p < r (d)q < r < p (e)r < p < q (f)r < q < p ConcepTest Section 4.7 Question 4

(c) Since p is a limit of the form 0/0 it can be evaluated by l’Hopital’s rule. The first application of the rule leads to another limit of the form 0/0, so l’Hopital’s rule must be applied a second time. Since q is a limit of the form ∞/∞ it can be evaluated by l’Hopital’s rule. ConcepTest Section 4.7 Answer 4 ANSWER

Since r is a limit of the form ∞*0 it can be rewritten in the form ∞/∞, and then l’Hopital’s rule can be applied. We have COMMENT: The limits for q and r could be the starting point for a discussion of dominance relations among logarithmic, power, and exponential functions. ConcepTest Section 4.7 Answer 4 ANSWER (cont’d)

ConcepTest Section 4.7 Question 5 Arrange in order by dominance as x  ∞, from least to most dominant.

ConcepTest Section 4.7 Answer 5 ANSWER (b), (d), (a), (e), (c). As x  ∞, in increasing order of dominance, we have x(ln x) 2 < x 2 < x 100 < e x lnx < e 2x / x. COMMENT: Do the students know the answer without using l’Hopital’s Rule?