Day 8, Mat128: Calculus A

Get yours before class!

• Answer the question. "What conclusions can you draw?"
• Show work.
• Give units.

• Basic rules:
  • Constant rule:

  • Identity rule:

• Sum law: "the limit of a sum is the sum of the limits"

  So we use the basic laws above, and this "limit law", to conclude that

  \[ \lim_{x\to{a}}(x+c)=a+c \]

• The Difference law, product law, quotient law (slight caveat here -- no zero denominators), power law, and root law work about the same way (remember the domain restriction for roots -- that is, fractional powers).

• As discussed last time, we can proceed to deduce that all polynomials have limits, and furthermore that

  \[ \lim_{x\rightarrow{a}}P(x)=P(a) \]

  Let's look at a simple example of the "polynomial rule": how can we show that it works for, say, $P(x)=9x^3-6x^2+x-7$?

  This is why it is nice to have a "polynomial law" -- so that we don't have to go through this every time.

• Other Laws:
  • The squeeze theorem What can we say about the limit of g as x approaches a?

• Now let's take a look at some of the examples from the section exercises.

  Keep your eye on some of the "tricks" we use to turn ugly indeterminate expressions into equivalent expressions for which we can evaluate the limits.

  • Example 3, p. 65
  • #7, p. 69
  • #10
  • #25
  • #33
  • #47
  • #54

• Additional examples:
  • #63, p. 71
  • #59, p. 71 (squeeze it!)