Real Analysis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 8 July 2014

Lecture 6

Graphing and limits

Graph f(x)= { sin(1x), ifx0, 1, ifx=0. x y 1 -1 -1π -12π 12π 1π 2π limxaf(x)= if f(x) gets closer and closer to as x gets closer and closer to a.

limx0sin(1x) does not exist because sin(1x) oscillates between 0 and 1 as x gets closer and closer to 0.

The function f(x) is continuous at x=a if f(x) is such that limxaf(x) =f(a). The function f(x)= { sin(1x), ifx0, 1, ifx=0 is not continuous at x=a because limx0f(x)=limx0sin(1x) does not exist.

Graph y=tanx=sinxcosx. - 2 π - 3 π 2 - π 2 π 2 π 3 π 2 2 π x -1 1 y Notes:

(a) If sinx=0 then y=0.
(b) If cosx=0 then y is very large.

limxπ2tanx does not exist because tanx gets larger and larger as x gets closer and closer to π2.

Graph y=x.

x=n, if n and nx<n+1, i.e. x is the largest integer x. - 3 - 2 - 1 1 2 3 4 x - 2 - 1 1 2 y limx1x does not exist because limx1- x=0and limx1+ x=1, where limx1-x is the limit of x as x gets closer and closer to 1 from the negative side of 1, and limx1+x is the limit of x as x gets closer and closer to 1 from the positive side of 1.

The function f(x)=x, for x, is continuous for x.

In English: A function f(x) is continuous at x=a if f(x) doesn't jump at x=a.

In math: A function f(x) is continuous at x=a if limxaf(x)=f(a).

Let z=12eiπ/4. Graph the sequence an=zn.

A sequence in is a function >0 n an The first 7 terms of this sequence are a1=12eiπ/4, a2=14ei2π/4, a3=18ei3π/4, a4=116ei4π/4, a5=132ei5π/4, a6=164ei6π/4, a7=1128ei7π/4. Graph x+iy as - 3 - 2 i 1 + 1 i x + iy 1 + 0 i x not i axis 0 + 1 i 0 + 2 i y i axis So a1= 12eiπ/4= 12(cosπ4+isinπ4)= 12(22+i22), a2- 14ei2π/4- 14eiπ/2= 14(cosπ2+isinπ2)= 14(0+i). a1 a2 a3 a4 a5 eiπ4 ei2π4 ei3π4 ei4π4 ei5π4 ei6π4 ei7π4 - 1 1 - i i limnan= means an gets closer and closer to as n gets larger and larger.

In our example the an are getting closer and closer to 0+0i, in a spiral.

Definition Let z=x+iy be in . The absolute value of z is |z|= x2+y2. In English: |z| is the distance from z to 0+0i.

The complex numbers is ={x+iy|x,y} withi2=-1.

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100312Lect6.pdf and was given on 11 March 2010.

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