In this segment, we'll talk about finding the limit of a function which is given in rational
form. So here, we have the question given to us. The statement is saying that hey you want
to find the limit of this particular expression as x approaches 5. If we look at the numerator,
at x equal to 5, your x squared minus 2x minus 15 will be equal to 5 squared minus 2 times 5
minus 15, and that gives us 0. And same thing for the denominator at x equal to 5. When we look at
x minus 5, that gives you 5 minus 5, which is 0. So what you are trying to find what you are
saying here is that you have a 0 over 0 form. And again, you can apply the L'Hopitals Rule to find
out what the limit is. So in this case, so what does L'Hopitals say? L'Hopitals says that if f(a)
is equal to 0, g(a) is equal to 0, then the limit of f(x) divided by g(x) as x approaches a can be
written as f'(a) divided by g'(a). So you could if both of those are defined, f'(a) as the g'(a). So
lets go ahead and see if that is the case. Here, what we have is that f(x) will be your x squared
minus 2x minus 15, and g(x) is your x minus 5. So let's go ahead and see if that is the case.
So what we have here, we have limit of x approaching 5, x squared minus 2x minus
15 divided by x minus 5. So we can say that hey it's the same equal to limit of x approaching 5
because we want to apply the L'Hopitals Rule to it. We can take the derivative of the numerator,
we can take the derivative of the denominator. And what is the derivative of the numerator? It
is 2x here and minus 2 here, so 2x minus 2. And of course derivative of 15 is 0. Derivative of x is 1
and derivative of 5 is 0, so that's what we get. So if we put now if we look at x approaching 5,
2x minus 2 has a finite value, 1 has a finite value. So we can just substitute
the value of 5 for x and be able to get what the limit is, in this case being 8.
So we applied the L'Hopitals Rule to be able to find out that the limit of this expression is 8.
Now, it's quite possible to do the same problem by possibly being able to factor the expressions.
So we have x squared minus 2x minus 15 divided by x minus 5. If we look at the top expression,
we can find the 0's of this function by using the quadratic formula. Or just by observation,
we can say that hey, I can rewrite the top the numerator by x minus 5 times x plus 3 and because
that gives me x squared minus 5x plus 3x and minus 15. And I can divide it by x minus 5. So this
and this cancels out. So this limit will be, as x approaches 5, it will be x plus 5. And since this
is a quantity which is finite at 5, so it will be 5 plus 3, and that gives us 8. So we're getting
the same limit as we got from L'Hopitals Rule as we're getting by factoring out the numerator
and canceling with the denominator as it's possible in this particular case. So either way,
you can be able to solve the problem of finding this limit. And that's the end of this segment.