Day 5, Mat129: Calculus I

1. wrap up section 1.6
2. start section 1.8

• We start with two especially trivial special limits:
  • Constant rule $(f(x)=c)$: $\lim\limits_{x\to{a}}c=c$

    Here the function $f(x)=1$ is plotted. As $x \to 0$, the $y$-values head to 1. Well, more to the point, they never vary from 1! Quite a boring function....

  • Identity rule $(f(x)=x)$: $\lim\limits_{x\to{a}}x=a$

    Here the function $f(x)=x$ is plotted. As $x \to 0$, the $y$-values head to 0. And the closer $x$ gets to 0, the closer $y$ gets to 0. That's the way it's supposed to work! Also a rather boring function: so predictable!

  What can we do with those? We need more horsepower....

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

  So what is this limit:

  $\lim\limits_{x\to{a}}(x+c)$

  This sum (and many other operations) satisfy the same pattern:

  "The limit of a STUFF is the STUFF of the limits."

  So, in particular,

  "The limit of a sum is the sum of the limits."
  $\lim\limits_{x\to{a}}(x+c)=\lim\limits_{x\to{a}}x+\lim\limits_{x\to{a}}c=a+c$

• Difference law
• Product law
• Constant multiple law:
• Quotient law (slight caveat here -- no zero denominators)

  Suppose that $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist. Then \[ \lim_{x \to a} [f(x)+g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \] \[ \lim_{x \to a} [f(x)-g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \] \[ \lim_{x \to a} [cf(x)] = c\lim_{x \to a} f(x) \] \[ \lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \lim_{x \to a} g(x) \] \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)} {\text{ if }} \lim_{x \to a} g(x) \ne 0 \]

• Power law \[ \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n \] where $n$ is a positive integer (the limit of a power is the power of the limit).

• Root law: \[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} \] where $n$ is a positive integer. Again, we can ``pass the limit inside''. Again, a caveat: why can't we compute

  \[ \lim\limits_{x\to{0}}\sqrt{x} \]

  Because the function isn't even defined as x approaches 0 from the left (square roots of negative numbers are imaginary). So we have only a one-sided limit, that \[ \lim\limits_{x\to{0^+}}\sqrt{x}=0 \] Of course we can say that \[ \lim\limits_{x\to{1}}\sqrt{x}=1 \] Limits exists everywhere else on the domain of the sqrt function.

• Putting these laws together
  • Now that we've got the sum and product and constant multiple rules, what can we deduce?
  • "Affine" rule, for one:
    \[ \lim\limits_{x\to{a}}(mx+b)=ma+b \]

  • Then we can proceed to deduce that all polynomials have limits, and furthermore that
    $\lim\limits_{x\rightarrow{a}}P(x)=P(a)$

    This is the conclusion of Exercise #55, p. 71, which says that polynomials are continuous, which we'll consider more in section 1.8.

    How do we know? Well an n^th degree polynomial can be written as a sum of constants $c_i$ multiplying powers of $x$, $x^i$: \[ P(x)=\sum_{i=0}^n c_ix^i \] Then the limit of the sum is the sum of the limits: \[ \lim_{x\to a}P(x)=\lim_{x\to a}\sum_{i=0}^n c_ix^i =\sum_{i=0}^n \lim_{x\to a}(c_ix^i) \] and then we use the constant multiple rule and the power rules, and we're done: \[ \lim_{x\to a}P(x) =\sum_{i=0}^n c_i\lim_{x\to a}x^i =\sum_{i=0}^n c_ia^i = P(a) \]

    Exercise #56 says it's also true for rational functions, for every element of their domain.

• Take a look at a couple of the examples in the text, examples 5 and 6 on page 66. These both illustrate the limiting process applied to the most important definition in calculus:
  $f'(x)=\lim\limits_{h\rightarrow{0}}{\frac{f(x+h)-f(x)}{h}}$

  In this case, both show examples of a slightly different version of this most important definition, the derivative at a point $x=a$:

  $f'(a)=\lim\limits_{h\to 0}{\frac{f(a+h)-f(a)}{h}}$

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

    What's happening to the $y$-values?

• Now let's take a look at some of the examples from the section exercises.
  • Example 3, p. 65
  • #7, p. 69
  • #10 (see the theorem, p. 65)
  • #25
  • #33
  • #47
  • #63, p. 71
  • #59, p. 71 (squeeze it!)

• A new administration in government?
• A sidewalk?
• Coffee poured into a cup?

Note: three things have to happen:

1. The limit of $f$ exists at $a$,
2. The function value $f(a)$ exists at $a$, and
3. The two values are equal.

Otherwise $f$ is discontinuous at $a$.

There are various kinds of discontinuity (which we've already seen):

• Removable discontinuities (e.g. holes)
• Jump discontinuities
• Infinite discontinuities
• "Crazy discontinuities" like the function of #59, p. 71: $x^2$ for rational values of $x$, and 0 for irrational ones.

This function has a limit at zero (-.5), but is not defined there. If $f$ is not defined at $x=0$, then it cannot be continuous there. We can fix this, by the way.... Just define $f(0)$ to be the limit (the closed dot in the graph above).

• We define one-sided continuity
• we have sum laws, product, and quotient laws. We've already used these to demonstrate that polynomials are continuous on all real numbers: \[ \lim_{x\to{a}}P(x)=P(a) \]

• Polynomials
• Rational functions: (ratios of polynomials)
• Trigonometric functions: sine, cosine, tangent
• Root functions: $f(x)=x^{1/n}$
• Exponential functions: $f(x)=b^x$

• Roughly: the composition of continuous functions is continuous.

• Let $F(x)=f(g(x))$ be a composite function.

  If $g$ is continous at $x=c$ and $f$ is continous at $g(c)$ then

  $F(x)$ is continuous at $x=c$.

• pp. 90-93, #3, 10, 12, 19, 23, 36, 46 (with Mathematica)
• That crazy function which was $x^2$ for rational values of $x$, and 0 for irrational ones, has an interesting property: it is continuous at only a single point! It satisfies the definition only at $x=0$: it's discontinuous everywhere else! That's pretty weird.

• Theorem: Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where $f(a)\ne{f(b)}$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.

• Example: pp. 93, #69