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# 1-sided vs. 2-sided limits (graphical)

Video transcript

So we have the graph of a
function right over here, and we want to think
about what does the limit of f of x as x
approaches 3 appear to be? And to do that, let's
just think about the limit as x approaches 3 from
values less than 3, and from values greater than 3. So let's first think
about the limit of f of x as x approaches
3 from values less than 3. So this little
negative superscript says we're going to
approach 3 from below 3. From 1, 2, 2.5, 2.99, 2.999. So if we approach-- so
this is 3 right over here. And we're going take the left
handed, or the left sided limit. We're going to approach 3
from this direction first. So when x is 0, f of x is there. When x is 1, f of x is there. When x is 2, f of x is there. When x is 2 and
1/2, f of x is at 5. When x is at-- looks like
roughly 2 and 3/4, we get to 4. Looks like about f
of x gets to 4.5. And so it looks like
as x approaches 3 from values less
than 3, it looks like our function
is approaching 4. So I would say it looks
like the left sided limit of f of x as
x approaches 3 is 4. Now let's do the same thing
for the right hand side. So the limit of f of x
as x approaches 3 from values larger than 3. So notice when x is equal
to 5, our f of x is up here. When x is equal to
4, f of x is here. When x is 3 and 1/2, it looks
like we're a little under 2 for f of x. And it looks like we're
getting closer and closer as x approaches 3 from
the positive direction, or from the right
side, it looks like f of x is getting closer
and closer to 1. So I would estimate,
based on this graph, that the limit of f
of x as x approaches 3 from the positive
direction is equal to 1. Now we have an issue. In order for this
limit to exist, we have to get the same value as
we approach from the left hand side and the right hand
side, but it's clear that we are not approaching
the same value when we go from the left
hand side as we do when we go from
the right hand side. So this limit right over
here does not exist. Does not exist. The only way that this
would have existed is if we got the same
value for both of these, and then the limit
would be that value. But we're clearly not
getting the same value.