L'Hopital's Rule solved questions

CHAPTER 25
L'Hopital's Rule

25.1 State L'Hopital's rule.
First, let us state the zero-over-zero case. Under certain simple conditions, if and
then Here, can be replaced by
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. and then
Here, again can be replaced by or The
conditions on / and g are the same as in the first case.

In Problems 25.2-25.53, evaluate the given limit.

25.2




25.3




25.4


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.


25.5




25.6

Here we have the difference of two functions that both approach °°. However,
which L'Hopital's rule is applicable.


25.7




208

, L'HOPITAL'S RULE 2Q9

25.8




25.9




25.10

to which L'Hopital's rule applies (zero-over-zero case).




25.11




25.12

to which L'Hopital's rule applies. (This result
was obtained in a different way in Problem 23.44.)

25.13

Then By Problem 25.12, in y=0. Hence,


25.14

(as in Problem 23.43).


25.15

Then By Problem 25.14, In y+0. Hence,



25.16




25.17

Note that L'Hopital's rule did not apply.