U0S1V09 The overall limit.txt

#
# File:   content-mit-18-01-1x-captions/U0S1V09 The overall limit.txt
#
# Captions for MITx 18.01.1x module [O2BEgM8kHoY]
#
# This file has 58 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

We've learned about left- and right-hand limits.
The left-hand limit is asking about what happens when
x approaches a from the left.
So it's concerned with values of x
that are over here to the left.
And the right-hand limit is talking
about values of x that are close to a,
but on the right over here.

Now a lot of times we just want to think
about values of x that are close to a, period,
without restricting to one side or the other.
And that is going to be the overall limit.
We'll denote it by limit of f of x as x approaches a.
So notice that there's no plus or minus sign here
and this will equal L, if whenever
x comes in close to a from either side
f of x gets really close to L.
In other words, the overall limit
equals L exactly when the left-hand limit
and the right-hand limit are both equal to the same number
L. In pictures it has to look like this.
As we come in from the left, f of x is approaching L.
And as we come in from the right, same thing.
And we can denote this either with this lim notation
or we can use arrows, f of x approaches L as x approaches a.
But remember, limits only care about values
of x that are close to a, but not equal to a.
So for the sake of the overall limit,
it's not going to matter whether we have a dot for f of a.
We could fill in the circle or we could put a dot down here
or we could just not have a dot if f of a doesn't exist,
whatever, it won't affect the limit.

Now the overall limit might not exist.
One way it might it fail to exist
is if the limit from one side does not exist.
So for instance as we come in from the left, f of x
might blow up to minus infinity or something.
But another way is if the limit from the left and the limit
from the right both exist, but they're not equal,
something like this.
They have to both exist and be equal in order
to get an overall limit.

So that's the overall limit.
Down the road will stop saying overall.
And just anytime we say the limit
without specifying left to right,
we'll mean this overall limit.
And this idea of the limit is really the building block
for all of calculus.
So we have some problems for you to get used to it
and then we'll start to develop quicker and more accurate
ways of computing these limits.
See you then.