MATH 221

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

Last updated: 13 August 2014

Lecture 13

Existence of limits

What is limx01x? If x=.1 then 1x=10.
If x=.01 then 1x=100.
If x=.001 then 1x=100.
If x=.0001 then 1x=100.
So, it looks like limx01x=.
If x=-.1 then 1x=-10.
If x=-.01 then 1x=-100.
If x=-.001 then 1x=-100.
If x=-.0001 then 1x=-100.
So, it looks like limx01x=-.
Since limx0+1x= and limx0-1x=, limx01x=UNDEFINED. y x -1 1 -1 1 f(x)=1x

limx-1lnx=???

Look at the graph of lnx.

x y 1 y=ex

Notes:
e0 = 1, e1 = 2.718
e2 8.8, e3 25
e-1 13, e-1 19

x y 1 y=lnx

Notes:
y=lnx means ey=x
So this graph is the same as the left one but with x and y switched.

So, from the graph, lnx doesn't even make sense for x close to -1. So limx-1lnx is certainly undefined.

Note: If we allow x to get closer and closer to -1 and be a complex number then ln-1=iπ andi3πand i5π since eiπ=cosπ+isinπ=-1+i·0=-1 and ln-1=iπ. Still limx-1lnx is undefined since it can't be iπ and 3iπ and 5iπ all at once.

limxsinx

The graph of sinx is x y 1 -1 y=sinx So, as x gets larger and larger, sinx keeps going back and forth between -1 and +1. So sinx doesn't get closer and closer to anything as x gets larger and larger. So limxsinx is undefined.

Continuous functions

A function is continuous if f(x) doesn't jump when x changes. The function f(x) is not continuous exactly at the places where it jumps.

A function f(x) is continuous at x=a if it doesn't jump at x=a, i.e. if limxaf(x)=f(a). a x y y=f(x) Not continuous at x=a.

f(x)=x is the round down function. 3.2=3.

-5 -4 -3 -2 1 2 3 4 5 x -4 -3 -2 -1 1 2 3 4 y f(x) is continuous if x0,±1,±2,±3,

Note: limx1-x=0 and limx1+x=1.

f(x)=x is the round up function. 3.2=4.

f(x)= { 1+x2, 0x1, 2-x, x>1. x y 1 2 3 1 2 y=f(x) f(x) jumps at x=1. limx1- f(x) = limx1- 1+x2=2. limx1+ f(a) = limx1+ 2-x=1. So limx1f(x) is UNDEFINED.

f(x)= { sin3xx, ifx0, 1, ifx=0. sin3x is continuous everywhere and x is continuous everywhere. So sin3xx is continuous everywhere. EXCEPT, it makes no sense when x=0. Now what is happening when x=0? limx0 sin3xx= limx0sin3x3x·3 =1·3=3. BUT f(0)=1, So limx0 f(x) f(0) in this case. So f(x) is not continuous when x=0. y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=sinx y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=sin3x x y y=1x x y y=sin3xx

Notes and References

These are a typed copy of Lecture 13 from a series of handwritten lecture notes for the class MATH 221 given on October 6, 2000.

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