
Applications
of differentiation  the graph of a function and its derivative 
L'Hospital's
rule  limits of indeterminate forms 
Applications
of L'Hospital's
rule  evaluation of limits of indeterminate forms, examples 





L'Hospital's
rule  limits of indeterminate forms 
The
rule, based on the generalized mean value theorem, that enables the
evaluation of the limit of an indeterminate
quotient of functions as the quotient of limits of their derivatives is
called L'Hospital's rule. 
a)
Indeterminate
forms 


Let
f
and g
are differentiable functions near a
(except possibly at a)
and g'(x) is
not
0,
and if 

or 


then 


The
rule can also be applied in case when a
= oo. 
If
the quotient of the derivatives repeats an indeterminate form
then we can proceed with the quotients of successive
derivatives, provided required conditions hold. 
Another
indeterminate
forms 
b)
Indeterminate form
0 · oo 
Write
the product f (x)
· g
(x),
where 

as
a quotient 


c)
Indeterminate form
oo

oo 
Rewrite
given difference f (x)

g
(x)
into the product 

and
first solve the indeterminate 

form 

If 

then,
write the above product as 


d)
Indeterminate (or indefinite) powers,
1^{oo},
0^{0}
and oo^{0}
we solve taking the natural logarithm of the given expression
by writing 
f (x)^{g
(x)}
= y
or
ln
y
=
g
(x)
·
ln
f (x) 
so
it then becomes an indeterminate product. 
There
are cases where we should combine both L'Hospital's rule and elementary
methods to evaluate limits. 

Applications of L'Hospital's
rule  evaluation of limits of indeterminate forms, examples 
Example:
Evaluate
the following limit 


Solution:




Example:
Evaluate
the following limit 


Solution:




Example:
Evaluate
the following limit 


Solution:




Example:
Evaluate
the following limit 


Solution:





Example:
Evaluate
the following limit 




Example:
Evaluate
the following limit 


Solution:




Example:
Evaluate
the following limit 


Solution:




Example:
Evaluate
the following limit 


Solution:





Example:
Evaluate
the following limit 


Solution:





Example:
Evaluate
the following limit 


Solution:












Calculus contents
D 



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