Applications of differentiation - the graph of a function and its derivative
L'Hospital's rule - limits of indeterminate forms
L'Hospital's rule and 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,  1oo00  and  oo0  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
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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 Example:   Evaluate the following limit
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