University:
Amherst College
Course:
MATH 111 | Introduction to the Calculus
Academic year:

2011
Views:

318

Pages:

2
Author:

Jacey Sweeney

Math 111, Fall 2011: Instructions for Limit Questions Here are some instructions to help you know what sort of answer I expect to a limit question. 1. If the question says something like “Use the precise definition of a limit to...” or “Use the /δ definition of a limit to...” then: • you must only use the precise definition • you cannot use the limit laws • you cannot use the graph • you cannot use any other informal reasoning such as: “as x approaches 0, ...” • your answer should have the following form: Take > 0 and let δ = ... (something in terms of ). Then 0 < |x − a| < δ =⇒ . . . =⇒ . . . ... =⇒ |f (x) − L| < . • you may also include your scratch work to help explain how you decided what δ should be, but that is not necessary for getting full credit. Example: Use the precise definition of a limit to show that lim (−3x + 1) = −2. x→1 Solution: Take > 0 and let δ = /3. Then 0 < |x − 1| < δ =⇒ =⇒ =⇒ =⇒ =⇒ |x − 1| < /3 3|x − 1| < |3x − 3| < | − 3x + 3| < (since |y| = | − y|) |(−3x + 1) − (−2)| < . 2. If the question says something like “Use the limit laws to...” then: • you may use any limits given in the question (or that you have calculated in earlier parts to the question), but no others • every step in your answer should use just one limit law and it should be clear from the way you write it which law you used. For example, writing lim (x + 3) = 2 + 3 = 5 x→2 is NOT enough. Instead you should write lim (x + 3) = (lim x) + (lim 3) = 2 + 3 = 5 x→2 x→2 x→2 and only if the question says you can assume that limx→a x = a and limx→a c = c. • you may not use the fact that any particular functions is continuous unless the question tells you that you can • if you use the limit laws for quotients or square-roots, make sure you explain why it is OK to do so (i.e. that the denominator is not zero, or the limit inside the square-root is not negative). Example: Suppose that lim f (x) = 2, x→3 lim g(x) = 3. x→3 2 (x) . Find limx→3 fg(x) Solution: Because limx→3 g(x) 6= 0, we have f (x)2 limx→3 f (x)2 (limx→3 f (x))(limx→3 f (x)) 2(2) 4 = = = = . x→3 g(x) limx→3 g(x) limx→3 g(x) 3 3 lim 3. If the question gives no particular instructions other than “Find the limit...” then: • you may use any of the facts or results given in class or the textbook, but you should refer to those that you use at key steps in your answer • you may condense the use of several different limit laws into one step and just say “by the limit laws” • you may use the fact that all of the standard functions we have looked at are continuous at every point in their domain Example: Find limx→2 Solution: We have 1 − 12 x x−2 . If x 6= 0, 2, we have 2−x − 12 2−x = 2x = x−2 x−2 2x(x − 2) 1 x 2−x −1 = . 2x(x − 2) 2x Therefore lim x→2 which, by limit laws, is equal to 1 x − 21 −1 = lim x − 2 x→2 2x −1 . 4

Math 111, Fall 2011 Instructions for Limit Questions

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