THEOREM (L'Hospital's Rule): Suppose and are differentiable and
near (except possibly at ). Suppose that

or that

(In other words, we have an indeterminate form of type or ) Then

if the limit on the right side exists (or is or ).

Indeterminate Forms of Type and

EXAMPLES:

1. Find

Solution 1: We have

Solution 2: We have

In short,

2. Find

Solution 1: We have

Solution 2: We have

3. Find

Solution: We have

In short,

4. Find

Solution: We have

5. Find

Solution: We have

6. Find

Solution 1: We have

Since
is an indeterminate form of type we can use L'Hospital's Rule again. But it
is easier to do trigonometry instead. Note that

Therefore

In short,

Solution 2??? (WRONG!): We have

Common Mistakes

7. Find

Solution 1: We have

It is easy to show that and by the Squeeze Theorem. Therefore

Solution 2(???): We have

One can show, however, that does not exist. In fact, we first note that
and may attain any value between and From this one can deduce that attains any nonnegative value infinitely
often as This means that does not exist, so L'Hospital's Rule can't
be applied here.

8. Find

Solution(???): We have

This is WRONG. In fact,
although the numerator as
notice that the denominator does not approach so L'Hospital's Rule can't be applied here.
The required limit is easy to find, because the function
is
continuous at and the denominator is nonzero here:

9. Find

Solution(???): We have

The answer is correct, but the solution
is WRONG. Indeed,
the above proof is based on the formula But this result was deduced from the fact
that
(see
). So, the
solution is wrong because it is
based on Circular Reasoning which is a logical fallacy.
However, one can apply L'Hospital's Rule to modifications
of this limit (see
).

Appendix B

1. Find

Solution 1: We have

Solution 2: We have

2. Find

Solution 1: We have

Solution 2: We have

3. Find

Solution 1: We have

Solution 2: We have

4. Find

Solution 1: We have

Solution 2: We have

Appendix A

THEOREM: The function is differentiable and

Proof: We have

Indeterminate Forms of Type
and

EXAMPLES:

10. Find

Solution 1: We have

Note that

therefore

Solution 2: We have

11. Find

Solution: We have

In short,

12. Find

Solution 1: We have

We can now proceed in two different ways. Either

or

In short,

Solution 2: We have

Indeterminate Forms of Type
and

13. Find

Solution (version 1): Note that is type of an indeterminate form. Put

then

We have

Therefore

Solution (version 2): We have

14. Find

Solution: Note that is
type of an indeterminate form.
Put

then

We have

Therefore

In short,

We have

Therefore

15. Find

Solution: Note that is type of an indeterminate form. Put

then

We have

Therefore

In short,

We have

Therefore

16. Find

Solution: Note that is type of an indeterminate form. Put

then

We have

Therefore

In short,

We have

Therefore

17. Find

Solution: Note that is type of an indeterminate form. Put

then

We have

Therefore

In short,

We have

Therefore

18. Find

Solution 1: Note that is type of an indeterminate form. Put
then

We have

Therefore

In short,

We have

Therefore

Solution 2: We have

Since it follows that Therefore

COMPARE: We have

since

Similarly,

since

19. Find

Solution: Note that is type of an indeterminate form. Put