SECTIONS 7.7 and 8.8 - FORMULA SHEET
7.7 - INDETERMINATE FORMS AND L’HOSPITAL’S RULE
Suppose fand gare differentiable.
•TYPE 0/0 If limx→af(x) = 0 and limx→ag(x) = 0 then
lim
x→a
f(x)
g(x)= lim
x→a
f0(x)
g0(x)
•TYPE ∞/∞If limx→af(x) = ±∞ and limx→ag(x) = ±∞ then
lim
x→a
f(x)
g(x)= lim
x→a
f0(x)
g0(x)
•TYPE 0· ∞ If limx→af(x) = 0 and limx→ag(x) = ±∞, then rewrite the product f g as a
quotient:
fg =f
1/g , or fg =g
1/g
to convert it to type 0/0 or type ∞/∞. Then use the L’Hospital’s Rule to evaluate the limit.
•TYPE ∞ − ∞ If limx→af(x) = ∞and limx→ag(x) = ∞, then convert the limit
limx→a[f(x)−g(x)]
into type 0/0 or ∞/∞by using a common denominator, rationalization, or factoring out a
common factor. Then use L’Hospital’s rule to evaluate the limit.
•TYPE 00,∞0,1∞Let y=f(x)g(x). Take ln of both sides, use properties of logarithms to
simplify, find the limit L, and finally take eLto get the answer.
NOTE THAT THESE ARE THE ONLY TYPES OF INDETERMINATE FORMS.
SECTION 8.8 IMPROPER INTEGRALS
•(a) R∞
af(x)dx = limt→∞ Rt
af(x)dx. The integral is called convergent if the limit exists and
divergent if it does not exist.
•(b) Rb
−∞ f(x)dx = limt→−∞ Rb
tf(x)dx. The integral is called convergent if the limit exists
and divergent if it does not exist.
•(c) R∞
∞f(x)dx =Ra
−∞ f(x)dx +R∞
af(x)dx for any real number a, provided both the integrals
are convergent.
•(d) If fis continuous on [a, b) but discontinuous at b, then Rb
af(x)dx = limt→b−Rt
af(x)dx.
•(e) If fis continuous on (a, b] but discontinuous at a, then Rb
af(x)dx = limt→a+Rb
tf(x)dx.
•(f) If fhas a discontinuity at c, where a < c < b then Rb
af(x)dx =Rc
af(x)dx +Rb
cf(x)dx,
provided both the limits on the right hand side exist and are finite.
EXAMPLE: The integral R∞
a
1
xpdx is convergent if p > 1 and divergent if p≤1.
