.L'Hopital's Rule: negative exponential function times x2: Find
.
If we were to substitute directly, we would have the form 
. Hence, we can't yet apply L'Hopital's Rule. If we rewrite the problem as
we now have the indeterminate form 
. Now we can apply L'Hopital's Rule. That is, we differentiate the numerator and the denominator separately, take the ratio, and evaluate the limit of the resulting expression. (We do not use the Quotient Rule).
Applying this rule, we get
Carrying out the differentiation, we get
. This is still in the form . We shall try applying L'Hopital's Rule again:
