- [Voiceover] Let's see if we can figure out the limit
of x over natural log of x as x approaches one.
And like always pause this video and see if you can
figure it out on your own.
Well we know from out limit properties this is going to
be the same thing as the limit
as x approaches one of x over
over
the limit,
the limit
as x approaches one
of the natural log
of x.
Now this top limit, the one I have in magenta,
this is pretty straight forward,
if we had the graph of y equals x
that would be continuous everywhere
it's defined for all real numbers and it's continuous
at all real numbers.
So it's continuous to limit as x approaches one of x.
It's just gonna be this evaluated x equals one.
So this is just going to be one.
We just put a one in for this x.
For the numerator here we just evaluate to a one.
And then the denominator,
natural log of x is not defined for all x's,
therefore it isn't continuous everywhere.
But it is continuous at x equals one.
And since it is continuous at x equals one,
then the limit here is just gonna be the natural log
evaluated at x equals one.
So this is just going to be the natural log
the natural log of one.
Which of course
is zero.
E to the zero power is one.
So this is all going to be equal to
this is going to be equal to
we just evaluate it
one
over
one
over
zero.
And now we face a bit of a conundrum.
One over zero is not defined.
It is was zero over zero, we wouldn't necessarily be
done yet but it's indeterminate form
as we will learn in the future there are tools we can apply
when we're trying to find limits and we evaluate it
like this and we get zero over zero.
But one over zero.
This is undefined
which tells us that this limit
does not exist.
So does
not
exist.
And
we are done.