You want to find the limit as x approaches
infinity of natural log two x divided by x.
But notice, in this case, as x approaches positive infinity
both the numerator and the denominator approach
positive infinity.
So we say this limit has an indeterminant form of
infinity divided by infinity.
And because of the indeterminant form,
we can use L'Hopital's Rule to help us determine this limit.
Where L'Hopital's Rule also called Bernoulli's Rule,
uses derivatives to evaluate limits involving
indeterminant forms.
Where zero divided by zero is probably the most
common indeterminant form,
if the limit has any of these indeterminant forms,
we can apply L'Hopital's Rule to determine the limit.
L'Hopital's Rule says
the limit as x approaches c of f of x divided by g of x,
is equal to the limit as x approaches c of f prime of x
divided by g prime of x.
So if this limit has one of these indeterminant forms,
we can find the derivative of the two functions
and then see if we can find the limit in this form here.
So going back to our example, let's try it.
This limit is equal to the limit as x approaches infinity
of the derivative of natural log two x
divided by the derivative of x.
Well the derivative of natural log two x would be
equal to one divided by two x times the derivative of two x
which would be two.
So we'd have one divided by two x times two,
the derivative of x is just one,
let's go and simplify this.
Two over two simplifies to one.
So we have one over x divided by one,
so we just had the limit as x approaches infinity
of one divided by x.
Notice in this form, the numerator stays at one,
and the denominator increases without bound,
and therefore this fraction gets smaller and smaller
and approaches zero, therefore this limit is equal to zero.
So using L'Hopital's Rule, we were able to
determine this limit.
Let's look at the graph of the original function,
see if they can verify this limit.
Here it is.
Notice as x approaches infinity,
or as I move right along the graph,
notice how the y values or function values,
are slowly approaching zero, verifyng our limit is correct.
Let's take a look at our second example.
Here we have the limit as x approaches infinity of
two x cubed divided by e raised to the power of four x.
Again notice, as x approaches positive infinity,
both the numerator and denominator approach
positive infinity.
And therefore we can apply L'Hopital's Rule to
help us determine this limit.
So we'd have the limit as x approaches infinity
of the derivative of the numerator divided by
the derivative of the denominator.
The derivative of two x to the third,
that would be six x squared.
The derivative of e raised to the power of four x
would be either four x times four or
four e to the four x.
Notice how, if we wanted to,
we could simplify the four and the six,
but there's no need to.
Notice how in this form, both the numerator
and the denominator are still approaching infinity,
which means we can apply L'Hopital's Rule again.
So this would be equal to the limit as x approaches infinity
of the derivative of the numerator divided by
the derivative of the denominator.
The derivative of six x squared would be 12 x.
The derivative of four e to the four x would be
four e to the four x times four or 16 e to the four x.
Once again, notice as x approaches infinity,
this is approaching infinity over infinity,
so we can apply L'Hopital's Rule again.
The derivative of 12 x would be 12.
The derivative of 16 e to the four x would be
16 e to the four x times four, or 64 e to the four x.
And now because the numerator is a constant,
we can determine this limit.
The limit as x approaches infinity of 12
divided by 64 e to the four x,
the numerator is the constant 12,
the denominator is increasing without bond,
and therefore this fraction is getting smaller and smaller,
approaching zero.
And therefore this limit is equal zero.
Again, let's verify this by looking at the graph of
the given function.
Here it is and notice how very quickly we can tell
the function values are approaching zero,
as we move to the right, or as x approaches infinity
in this direction here.
I hope you found this helpful.