L ' Hospital's Rule

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Suppose that and .

 

Recall that if , then  .

 

If but , then .

 

For the case where both and are 0, we are not sure whether the limit exists or not. In fact, although the limit exists, it is not obvious what is its value. In this case, is called an indeterminate form of type .

 

Let's study the following three easy examples first :

Example 1 , , .

Example 2 , , does not exist.

Example 3 , , .

 

Since f and g are differentiable at 0, the graphs of f and g resemble the tangent lines of and at (), respectively. The derivative of at 0 equals 0 and the derivative of at 0 equals 1, we can see that the graphs of f and g look very much like the lines and .  It is not surprised that .

 

Suppose now we want to find , let's plot and on the same screen for x near 1.

 

For x near 1, the graphs of and both look like the straight line , it is reasonable to guess that . The values of for x near 1 also confirm our guess.

 

f:=x->ln(x)/(x-1):
for n from 1 to 9
do
x:=1.1-n/100:

 

n:='n':   x:='x':
for n from 1 to 9
do
x:=0.9+n/100:

 

Intuitively, if x is sufficiently close to 1, then the value of is close to

+ f ' (1) ( ) =

Hence,

= .

 

Another case where a limit is not obvious is the limit of the form

where both and equals or and it is called an indeterminate form of type .   For instance, .

 

From the plot above, we see that when x is sufficiently large the graph of looks like a constant function. Since

,

for a sufficiently large the slope of the tangent line of at is almost 0.

We would expect that

.

We can also see this by observing the values of for x large.

f:=x->ln(x)/(x-1):
for n from 1 to 6
do
x:=10^(n+3):

 

The examples above seem to suggest that we can find the limit of those indeterminate forms by the limit .

 

L'Hospital's Rule:

Suppose that f and g are differentiable and g ' ( x ) does not equal to 0 near a , except possibly at a . Suppose that

and

or and or

Then

.

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

 

Remarks :

1. L'Hospital's Rule is also valid for one-sided limits and for limit at infinity or negative infinity.

2. For the special case in which , and ,  f ' and g ' are continuous, and , it is not hard to see why l ' Hospital ' s Rule is true.

 

 

=

 

=

 

=

 

=

 

Geometrically, as we have seen in those examples above, the graphs of f and g look like their tangent at a if we zoom in towards the point ( ). The equations of the tangent lines of and are

y = f( a ) + f ' ( a ) ( ) = f ' ( a ) ( )

and

y = g( a ) + g ' ( a ) ( ) = g ' ( a ) ( ),

respectively.

This suggests visual that

= .

Example: .

 

Since

and ,

 

We can apply l' Hospital ' s Rule :

and

Since

,

 

we have

= 1.

 

 

Example: .

Since

and ,

 

and ,

 

and

.

 

By l ' Hospital ' s Rule, we get

.

 

 

Example: .

 

Since

and ,

 

= and ( ) = ,

and

.

By l ' Hospital ' s Rule, we get

.

There are situations for which we have to use l ' Hospital ' s Rule more than once. For instance,

Example: .

Note that

and .

 

= and .

 we have again is an indeterminate form of type , since

= and .

By l ' Hospital ' s Rule, we get

= 0,

and hence,

= 0.

 

Other type of indeterminate forms

 

(I). If and (or ), the limit is called an indeterminate form of type . We can deal with it by writing the product f g as a quotient :

or

 

Example: .

Note that

and .

Writing , we have , so l ' Hospital ' s Rule gives

 

= = = 0

 

Note that another option is to write

 

 

This gives an indeterminate form of type , but if we apply l ' Hospital ' s Rule we get a more complicated expression than we started with. In general, we should choose the option that leads to simpler limit.

(II). If and , the limit is called an indeterminate form of type . We shall try to convert the difference into a quotient so that we have an indeterminate form of type (or ).

 

Example: .

 

Note that

and .

 

Since

 

and

and .

 

So we an indeterminate form of type , l ' Hospital ' s Rule gives

= 0.

 

(III). Indeterminate Powers

(a) If and , the limit is called an indeterminate form of type .

 

(b). If and , the limit is called an indeterminate form of type .

 

(c). If and (or ), the limit is called an indeterminate form of type .

 

Each of these three cases can be treated by writing the function as an exponential :

 

 

Example: .

 

Note that this limit is an indeterminate form of type , and we can write as .

We have shown that

,

 

since is continuous at 0, we get

= = 1.

 

 

Problems to think about :

1. Show that , for all positive integer n . What does this imply ?

 

2. What's wrong with the argument below :

"  (again an indeterminate form of type )

 

= which does not exist !

 

Hence, does not exist. "