There is an interesting rule which is often useful for evaluating difficult limits called L’Hôpital’s^{2} rule. The best versions of this rule are based on the Cauchy Mean value theorem, Theorem 5.9.2 on Page 376.
 (5.11) 
and f^{′} and g^{′} exist on
 (5.12) 
Then
 (5.13) 
Proof: By the definition of limit and 5.12 there exists c < b such that if t > c, then

Now pick x,y such that c < x < y < b. By the Cauchy mean value theorem, there exists t ∈

Since g^{′}

and so, since t > c,

Now letting y → b−,

Since ε > 0 is arbitrary, this shows 5.13.
The following corollary is proved in the same way.
 (5.14) 
and f^{′} and g^{′} exist on
 (5.15) 
Then
 (5.16) 
Here is a simple example which illustrates the use of this rule.
Example 5.15.3 Find lim_{x→0}
The conditions of L’Hôpital’s rule are satisfied because the numerator and denominator both converge to 0 and the derivative of the denominator is nonzero for x close to 0. Therefore, if the limit of the quotient of the derivatives exists, it will equal the limit of the original function. Thus,

Sometimes you have to use L’Hôpital’s rule more than once.
Example 5.15.4 Find lim_{x→0}
Note that lim_{x→0}
Warning 5.15.5 Be sure to check the assumptions of L’Hôpital’s rule before using it.
Example 5.15.6 Find lim_{x→0+}
The numerator becomes close to 1 and the denominator gets close to 0. Therefore, the assumptions of L’Hôpital’s rule do not hold and so it does not apply. In fact there is no limit unless you define the limit to equal +∞. Now lets try to use the conclusion of L’Hôpital’s rule even though the conditions for using this rule are not verified. Take the derivative of the numerator and the denominator which yields
Some people get the unfortunate idea that one can find limits by doing experiments with a calculator. If the limit is taken as x gets close to 0, these people think one can find the limit by evaluating the function at values of x which are closer and closer to 0. Theoretically, this should work although you have no way of knowing how small you need to take x to get a good estimate of the limit. In practice, the procedure may fail miserably.
Example 5.15.7 Find lim_{x→0}
This limit equals lim_{y→0}
There is another form of L’Hôpital’s rule in which lim_{x→b−}f
 (5.17) 
and f^{′} and g^{′} exist on
 (5.18) 
Then
 (5.19) 
Proof: By the definition of limit and 5.18 there exists c < b such that if t > c, then

Now pick x,y such that c < x < y < b. By the Cauchy mean value theorem, there exists t ∈

Since g^{′}

and so, since t > c,

Now this implies

where for all y large enough, both

Therefore, for y large enough,

due to the assumption 5.17 which implies

Therefore, whenever y is large enough,

and this is what is meant by 5.19. ■
As before, there is no essential difference between the proof in the case where x → b− and the proof when x → a+. This observation is stated as the next corollary.
 (5.20) 
and f^{′} and g^{′} exist on
 (5.21) 
Then
 (5.22) 
Theorems 5.15.1 5.15.8 and Corollaries 5.15.2 and 5.15.9 will be referred to as L’Hôpital’s rule from now on. Theorem 5.15.1 and Corollary 5.15.2 involve the notion of indeterminate forms of the form
It is good to first see why this is called an indeterminate form. One might think that as y →∞, it follows x∕y → 0 and so 1 +

Now using L’Hôpital’s rule,

Since exp is continuous, it follows
