In Lecture: More §2.2/2.3 (Limits) Office hours today: 2:00-3:00 in my office, PDL C-326

Today:

Read Textbook 2.3-2.5○WORK ON Hwk03 & Hwk04 (2.2 & 2.3), closing today & Fri○

To Do:

Quiz tomorrow, 20 minutes, based on the first 3 homeworks (tangents to circles, graphical limits, parametric equations)

Ask in quiz section tomorrow ○Post a discussion/question in Canvas○

Come to Office Hours○Math Study Center (in CMU B-014)○

Have Questions?

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Every good calculus student must know her limits! *rimshot*

Wednesday, April 8

Meaning: the limit of a quotient

, as the dominator is _____

Meaning: as gets smaller but remains positive, the reciprocal values

get: ______________________________________

Meaning: as gets smaller but remains negative, the reciprocal values

get: ______________________________________

Meaning: as gets larger and larger, the reciprocal values

get __________________ (note: here the sign does not matter)

A good example that includes all three situations is the function

Other examples:

Some important limits to know, in shorthand notation:

Let be a constant

8) SUBSTITUTION LAW (consequence of all of the above):

Assume

and

both exist and are finite. Then the following hold:

.LIMIT LAWS (§2.3, see textbook):

8) SUBSTITUTION LAW (consequence of all of the above):

If is an expression in involving sums/powers/products/ratios which is defined at , then

Examples:

d) Determine the vertical asymptotes of the function:

use algebraic methods to simplify, factor/cancel terms, or rationalize, until the limit can be determined.

3) If we get an indeterminate expression, such as etc:

4) Try the Squeeze Thm (after this)

(Later this term we'll add: 5) Use L’Hospital; explained in chapter 4.4)

2) If dividing a non-zero number by 0, check the sign of the denominator. Depending on the sign, we get either , or if the sign of the denominator is different for the one sided limits, then the overall limit does not exist.

1) Always try substitution first. If the limit is computable by substitution, we're done. Else try the steps below.

(if time permits -- else we'll discuss this on Wednesday)General strategy for COMPUTING LIMITS:

Ex 1: If for all values of near 1, evaluate

Ex 2: Compute

IF: for all values of x in an interval containing ,

AND:

THEN:

too

The Sandwich (or Squeeze) Theorem (§2.3)