CHAPTER 25
L'Hopital's Rule
The
In Problems 25.2-25.53, evaluate the given limit.
25.2
25 .3
25.4
25 . 5
25.6
208
25.1
State L'Hopital's rule.
First, let us state the zero-over-zero case. Under certain simple conditions, if
then
Here,
can be replaced by
and
The conditions are that/and g are differentiable in an open interval around b and that g' is not zero in
that interval, except possibly at 6. (In the case of one-sided limits, the interval can have b as an endpoint. In
the case of *-» ±», the conditions on / and g hold for sufficiently large, or sufficiently small, values of x.)
The second case is the infinity-over-infinity case.
Here, again
conditions on / and g are the same as in the first case.
can be replaced by
and
then
Here, we have applied L'Hopital's rule twice in
succession. In subsequent problems, successive use of L'Hopital's rule will be made without explicit mention.
Here we have the difference of two functions that both approach °°. However,
which L'Hopital's rule is applicable.
25.7
or