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[MUSIC] Up to now, we've been thinking a 
lot about limits. 

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We've been considering f(x) when x is 
close to but not equal to a. 

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Let's refine that a bit now. 
Instead of thinking about x close to a. 

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I'm going to think about x close to the 
right side of a or x close to the left 

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side of a. 
Super-sized definition. 

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So we say the limit of f(x) as x 
approaches a from the right, I'm going to 

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use this little + sign to mean from the 
right. 

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So if this is equal to l, it means that 
f(x) is as close as you want to l, this 

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is just like before, provided x is near 
enough a on the right-hand side. 

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So we're going to use a little + sign to 
denote approaching from the right-hand 

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side. 
Now we're going to play a same game for 

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approaching from the left-hand side. 
So let's say the limit of f(x) as x 

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approaches a from the left-hand side is 
equal to l means that f(x) is as close as 

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you want to l just like usual, provided x 
is near enough a on the left-hand side. 

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Let's go see some picture of this at the 
blackboard. 

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Really helps to see a graph. 
Here's a graph of a made up function that 

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I'm called f(x). 
You'll note that f(x) has some issues at 

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the input 3. 
There's an empty circle here and a 

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filled-in circle here so if I plug in 3, 
my output valley is 2. 

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And indeed if I plug in numbers that are 
just a little bit above 3, I get out 

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numbers that are close to 2. 
On the other hand, if I plug numbers that 

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are a little bit less than 3, I get out 
numbers that are close to 1. 

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We can summarize our observations with 
these two statements. 

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The first is that the limit of f(x) as x 
approaches three from the right-hand side 

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is equal to 2. 
And that makes sense because I can get 

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the output of the function to be as close 
to 2 as I'd like if I'm willing to 

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evaluate the function of inputs that are 
close to but just a little bit bigger 

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than 3. 
Similarly, the limit of f(x) as x 

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approaches two from the left-hand side is 
equal to 1. 

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We think back in the graph, I was getting 
outputs that are close to one if I 

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evaluated the function inputs which are 
close to, but just a little bit less, 

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than 3. 
So in this case, the two sided limit 

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doesn't exist. 
I can't say that f(x) is getting close to 

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anything if all I know is that x is close 
to 3 but the left and the right hand 

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limits do exist. I can say that f(x) is 
getting close to 1. If x approaches 3 

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from the left-hand side, and I can say 
that f(x) is getting close to 2 if x 

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approaches 3 from the right-hand side. 
This situation comes up quite a bit where 

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you compute and you find that the right 
and the left hand limits exist but they 

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disagree and consequently, the two side 
of limit doesn't exist. 

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I can summarize it like this. 
If the limit of f(x) as x approaches a 

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from the right, is different than the 
limit of f(x) as x approaches a from the 

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left, then the two sided limit the limit 
of f(x) as x approaches a does not exist. 

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It works the other way as well. 
If the limit from the right-hand side is 

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equal to the limit from the left-hand 
side, let's call our common value l, then 

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the two sided limit, the limit of f(x) as 
x approaches a, no + or -. 

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So this is just the usual old limit. 
Then this limit exists and it's equal to 

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that same common value, l. 
The homework will challenge you with many 

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more situations of one sided limits. 
If you get stuck, contact us. 

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We are here to help you succeed. 
[MUSIC]