L’Hopital’s Rule. Example Problem
Let us consider the following example of L’Hopital’s rule.
If you evaluate the cosine of 0, you get 1. The exponential of 0 is 1, divided by 0 minus 0 – we get 0/0.
Factoring probably won’t work, since there are no algebraic methods or anything that will help us. We’ll use L’Hopital’s rule instead.
Thus, we take the derivative of the numerator (3sin3x), times 3 because of the chain rule, but you already knew that.
The derivative of e to the 4x is equal to 4 times e to the 4x divided by 2 times x minus 3.
In other words, we can take the limit of each factor separately, then combine them when we have a limit on both sides.
So you plug in zero for the x-value. The sine of 0 is 0, minus e to the 0 is 1, times 4 is -4, divided by 0 minus 3 is -3. So the limit would be 4/3. Here it is: L’Hopital’s rule.
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