7-5-65 L'Hopital's
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If we put in X approaching zero from the right, we end up with
zero times Infinity, which is one of the indeterminate forms.
But in order to use Lobital's rule, we have to have it as a
ratio.
So let's leave the 3X in the top and let's take tangent into the
bottom, which would give us cotangent Pi halves minus X.
Now if we look at putting 0 in, we get zero on the top and the
cotangent of π halves -0 is 0.
So this is an indeterminate form.
If we take the first derivative of the top and the bottom, we
get three over the derivative of cotangent is negative cosecant
squared Pi halves minus X.
But then we have to take the derivative of the inside.
The derivative of π halves is 0, but negative X is -1.
So we'd get the limit as X goes to zero.
From the right hand side, the negative and the negative are
going to cancel.
We get three over cosecant squared Pi halves minus X.
If I put in zero now from the right hand side, the cosecant of
π halves is just one, so we get 3 / 1 ^2 or three.