TR3.5

Limits

Houjun Liu 2021-09-27 Mon 12:00

1 Limits

1.1 Warming up

Here's a function

\(y = \frac{1}{x}\).

We know that it has

  • Domain \(D (-\infty, 0)(0, \infty)\)
  • Range \(R (-\infty, 0)(0, \infty)\)
  • \(As\ x\to\infty,\ y\to0\)
  • Function is odd, that is, \(f(-x) = -f(x)\)

1.2 The Limit Notation

1.3 Computing Limits Algebraically

1.4 Types of Discontinuity

1.5 Error and Epsilon Delta Proofs

1.6 CN10062020 Continuity

#disorganized #flo

\(\lim_{x \to a} f(x) \neq f(a)\).

Sometimes

*A function is continuous at \(x=a\) if ALL OF the following three conditions:$

  1. \(\lim_{x\to a} f(x)\) exists
  2. \(f(a)\) exists
  3. \(\lim_{x\to a} f(x) = f(a)\)

threestepslimit.png

Figure 1: threestepslimit.png

\definition{Removable discontinuity}{Removeable discontinuity are often holes. They are discontinuities that, with an additional definition, one could remove.} For instance, \(f(x) = \frac{x^2-x-2}{x-2}\) has a hole at \(x=2\), but if we defined a value for \(x=2\), our lovely discontinuity is immediately removed.

\defintion{Infitinite discontinuity}{Functions that approch infinity} If you think about it, if you try to fix the discontinuity, you will be tracing all the way up the infinity

\definition{Jump discontinuity}{"Staircase" functions that causes jump} Like…

jumpdisc.png

Figure 2: jumpdisc.png

As you could see, if you try to fix the discontinuity, this would result in vertical lines, which is illegal in functions.

Continuous-from-right: \(f(a) = \lim{x \to a^+} f(x)\) Continuous-from-left: \(f(a) = \lim{x \to a^-} f(x)\)