Last, determine the limit
using L'Hopital's Rule.
Let's first check the form of the limit.
As x approaches zero from the right
four divided by x approaches infinity.
And as x approaches zero from the right
four divided by sine x also approaches infinity.
Right now, the limit is in the form of infinity
minus infinity
which isn't indeterminate form
but it is not one of the indeterminate forms
where we can apply L'Hopital's Rule.
To apply L'Hopital's Rule,
we need to have the limit
in one of these indeterminate forms here in fraction form.
If we have one of these forms,
then L'Hopital's Rule states the limit as x approaches c
if f of x divided by g of x
equals a limit as x approaches c of f prime of x
divided by a g prime of x.
So just because we don't have
one of the correct indeterminate forms for L'Hopital's Rule,
it doesn't mean we can't use L'Hopital's Rule,
it just means we have to change the form
of the original function.
So in this case, let's try determining the difference
of four divided by x and four divided by sine x.
And let's work this out below.
We have four divided by x minus four divided by sine x.
To find the difference,
we need to obtain the least common denominator
which is the product of the two denominators or x sine x.
And therefore to find the difference
we need to multiply the numerator
and denominator of four over x by sine x.
And we need to multiply the numerator and denominator
of four over sine x by x.
Notice now we do have a common denominator of x sine x
and the numerator is now four sine x minus four x.
Which means a given limit is equal to the limit
as x approaches zero from the right of four sine x
minus four x divided by x sine x.
And now I'll check the form of this limit.
As x approaches zero from the right,
four sine x minus four x approaches zero minus zero or zero,
and as x approaches zero from the right
x times sine x approaches times zero which is also zero.
So notice, by determining the difference,
we now have a limit in one of the indeterminate forms
where we can apply L'Hopital's Rule
to help us determine the limit.
Applying L'Hopital's Rule,
this limit is equal to the limit
as x approaches zero from the right of
the derivative of four sine x minus four x
divided by the derivative of x sine x.
The derivative of four sine x is equal to four cosine x
minus the derivative of four x, which is equal to four.
And now we divide by the derivative of x sine x
which requires the product rule
where we have the first function of x
times derivative of the second function
which is a derivative of sine x, which gives us cosine x
and then plus the second function of sine x
times derivative of the first function
and the derivative of x is equal to one.
And now I'll check the form of this limit.
As x approaches zero from the right,
four times cosine x approaches four times one of four,
and then we have minus four,
the numerator approaches zero
and the denominator as x approaches zero from the right
x times cosine x approaches zero times cosine zero
or zero times one which is zero
and as x approaches zero from the right
sine x also approaches zero.
The limit is still in the form of zero divided by zero
which means we can apply L'Hopital's Rule again.
So this limit is equal to the limit
as x approaches zero from the right
of the derivative of four cosine x minus four
divided by the derivative of x cosine x plus sine x.
The derivative of four cosine x
is equal to four times negative sine x
or negative four sine x
and then minus a derivative of four, which is equal to zero.
And now in the denominator
to differentiate x times cosine x,
we need to apply that product rule
where we have the first function of x
times derivative of the second function
and the derivative of cosine x is equal to negative sine x.
And then we have plus the second function of cosine x
times the derivative of the first function
and the derivative of x is equal to one.
And then we have plus the derivative of sine x
which is equal to cosine x.
And now because we have cosine x
plus cosine x in the denominator,
I think we can now determine the limit
by performing direct substitution.
Let's go ahead and try.
As x approaches zero from the right,
negative four times sine x
approaches negative four times zero.
In the denominator,
as x approaches zero from the right
x times negative sine x approaches zero times zero or zero.
Now we have plus as x approaches zero from the right,
cosine x approaches one here as well as here
giving us one plus one.
Simplifying, we have zero divided by two,
which is equal to zero.
So after all that work we now know
the original limit is equal to zero.
And before we go, let's verify this graphically.
So here is the graph of the original function
f of x equals four divided by x minus four
divided by sine x.
And notice as we approach x equal zero
from the right or positive side,
we can see, we are approaching the y value
or function value of zero, verifying our limit.
I hope you found this helpful.