
8-6 Calculus – Differentiation and Integration
Note: If f ′(a) = f ′′(a) = f ′′′(a) = ¼ = f
n-1
(a) = 0 and g′(a) = g′′(a) = g′′′(a) = ¼ = g
n-1
(a) = 0.
But f
n
(a) and g
n
(a) are not both zero; then, by repeated application of L’ Hospital’s rule, we have
lim
( )
( )
lim
( )
( )
.
x a x a
n
n
f x
g x
f x
g x
→ →
−
−
=
1
1
Working Rule
If the limit of f(x)/g(x) as x tends to a takes the form 0/0, differentiate the numerator and denominator
separately w.r.t x and then take the limit to obtain lim
( )
( )
.
x a
f ' x
g' x
®
If the resultant expression also takes the
indeterminate form 0/0 as x tends to a, then repeat the above process until indeterminateness desists.
Example ...