Welcome to two examples
of determining limits using L'Hopital's Rule.
L'Hopital's Rule, also called Bernoulli's rule,
uses derivatives to evaluate limits
involving indeterminant forms.
The most common indeterminant form is zero divided by zero,
but these are also indeterminant forms.
L'Hopital's Rule states 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, if this limit here
involves an indeterminant form.
So, going back to our two examples,
notice as X approaches zero, two tangent X approaches zero,
and so does five X, and therefore,
this is in indeterminant form,
and we can apply L'Hopital's rule, but before we do this,
let's take a look at the graph of this quotient function.
Here it is.
Notice, as we approach X equals zero from the right,
and from the left,
even though the function is undefined at X equals zero,
we can see this limit does exist,
so we'll go ahead and apply L'Hopital's Rule,
which means this limit is equal to the limit
as X approaches zero, of the derivative of two tangent X,
divided by the derivative of five X.
I have listed our derivative formulas here, for reference.
Notice how these first two do include the chain rule,
but for tangent X, since the inner function would be X,
and the derivative of X would be one,
we don't have to apply the chain rule,
so the derivative of two tangent X
is equal to two secant-squared X,
and the derivative of five X would just be five.
Notice how we no longer have an indeterminant form,
and we can actually determine this limit
by performing direct substitution,
which means, this limit is equal to
two times secant-squared zero,
divided by five.
Remember, the secant function
is the reciprocal of the cosine function,
so if we find cosine zero, using the Unit Circle,
we can also determine secant zero.
Well, here's our Unit Circle.
Here's zero radians.
Remember, the X coordinate is equal to cosine theta,
so if cosine zero is equal to one,
well, the reciprocal of one is also one,
therefore secant zero is equal to one,
which means, this is equal to two times one-squared,
since we have secant-squared zero, divided by five,
which would just be two-fifths.
So then, go back and take a look
at the graph of our function just for a moment.
Notice how it does appear that as X approaches zero,
from the left and the right,
the function value is approaching two-fifths.
Looking at our second example,
we have the limit as X approaches zero of sine four X
divided by eight X.
Again, notice as X approaches zero,
both the numerator and denominator approach zero,
so we have an indeterminant form,
and therefore, we can apply L'Hopital's Rule.
So, this limit is equal to the limit
as X approaches zero of the derivative of sine four X
divided by the derivative of eight X.
Well notice, to find the derivative of sine four X,
we have a composite function,
so we do have to apply the chain rule,
where the inner function U would be equal to four X,
so the derivative of sine four X is equal to cosine four X
times the derivative of four X, which is four,
so notice how we have the derivative of the outer function
times the derivative of the inner function,
and we'll divide this by the derivative of eight X,
which is eight.
Notice, we no longer have an indeterminant form,
and again, we can find this limit
by performing direct substitution.
So, this limit is equal to,
I'm going to go ahead and put the four first,
four times cosine of, four times zero is zero,
divided by eight.
Well, cosine zero was equal to one, so we have
four divided by eight, which is equal to one-half.
Okay, I hope you found these two examples helpful.
We'll take a look at two more examples
of L'Hopital's Rule in the next video.