Finding the Limit

Date: 10/24/95 at 21:30:14
From: Anonymous
Subject: Definition of limit

For a limit to exist, the right side and left side have to equal 
each other. Say you have a limit such as "lim as x->4 for function 
(3x-4). To see if this function exists, you have to check it on 
both sides.  This is where my question is.  Would you plug in 3 
for the left and 5 for the right? If I do that I would not get the 
same answer.  So what am I doing wrong? I have asked my Calculus 
teacher and she said that was not it but she did not explain how. 
I have looked in several calculus books to try to teach myself, 
but the examples are not that good. Please help.

Date: 11/5/95 at 16:1:2
From: Doctor Jeremy
Subject: Re: Definition of limit

The thing that you're confused about is what "left" and "right" 
mean there.  They are a special kind of limit.  The "left" limit 
is the limit as you get closer to 4 from the left (below).  The 
"right" limit is the limit as you get closer to 4 from the right 
(above).  You want those limits to be the same or the limit 
doesn't really exist.

In this case, the limit on the left is 8:

	if x=3 then (3x-4)=5
	if x=3.9 then (3x-4)=7.7
	if x=3.99 then (3x-4)=7.97

The limit on the right is also 8:
	if x=5 then (3x-4)=11
	if x=4.1 then (3x-4)=8.3
	if x=4.01 then (3x-4)=8.03

So it gets closer to 8 as x gets closer to 4 on _either side_, 
meaning the limit exists and is 8.

Sometimes you can get right and left limits, but they don't match 
up so you don't have a limit.  Here's an example:

Let f(x) be 1 if x is greater than or equal to zero, and f(x) be 0 
if x is less than zero.  Then the limit of f(x) as x->0 on the 
left is 0, since whenever x is negative, no matter how close to 
zero, f(x) is 0.  But the limit on the right is 1, since whenever 
x is positive, no matter how small, f(x) is 1.  So lim as x->0 
f(x) doesn't exist, since the left and right limits don't match 
up.

But there are no particular numbers you can pick (such as you 
tried with 3 and 5) to get the "left" and "right".

-Doctor Jeremy,  The Math Forum