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Indeterminate Forms and L'hospital's Rules, Cheat Sheet of Calculus

Typology: Cheat Sheet

2020/2021

Uploaded on 04/23/2021

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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 limxaf(x) = 0 and limxag(x) = 0 then
lim
xa
f(x)
g(x)= lim
xa
f0(x)
g0(x)
TYPE /If limxaf(x) = ±∞ and limxag(x) = ±∞ then
lim
xa
f(x)
g(x)= lim
xa
f0(x)
g0(x)
TYPE 0· If limxaf(x) = 0 and limxag(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 limxaf(x) = and limxag(x) = , then convert the limit
limxa[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 00,0,1Let 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 = limtbRt
af(x)dx.
(e) If fis continuous on (a, b] but discontinuous at a, then Rb
af(x)dx = limta+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 p1.

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SECTIONS 7.7 and 8.8 - FORMULA SHEET

7.7 - INDETERMINATE FORMS AND L’HOSPITAL’S RULE

Suppose f and g are differentiable.

  • TYPE 0/0 If limx→a f (x) = 0 and limx→a g(x) = 0 then

lim x→a

f (x) g(x)

= lim x→a

f ′(x) g′(x)

  • TYPE ∞/∞ If limx→a f (x) = ±∞ and limx→a g(x) = ±∞ then

lim x→a

f (x) g(x)

= lim x→a

f ′(x) g′(x)

  • TYPE 0 · ∞ If limx→a f (x) = 0 and limx→a g(x) = ±∞, then rewrite the product f g as a quotient: f g =

f 1 /g

, or f g =

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→a f (x) = ∞ and limx→a g(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 eL^ to get the answer.

NOTE THAT THESE ARE THE ONLY TYPES OF INDETERMINATE FORMS.

SECTION 8.8 IMPROPER INTEGRALS

  • (a)

a f^ (x)dx^ = limt→∞

∫ (^) t a f^ (x)dx. The integral is called^ convergent^ if the limit exists and divergent if it does not exist.

  • (b)

∫ (^) b −∞ f^ (x)dx^ = limt→−∞

∫ (^) b t f^ (x)dx.^ The integral is called^ convergent^ if the limit exists and divergent if it does not exist.

  • (c)

∞ f^ (x)dx^ =^

∫ (^) a −∞ f^ (x)dx^ +^

a f^ (x)dx^ for any real number^ a, provided both the integrals are convergent.

  • (d) If f is continuous on [a, b) but discontinuous at b, then

∫ (^) b a f^ (x)dx^ = limt→b−

∫ (^) t a f^ (x)dx.

  • (e) If f is continuous on (a, b] but discontinuous at a, then

∫ (^) b a f^ (x)dx^ = limt→a+

∫ (^) b t f^ (x)dx.

  • (f) If f has a discontinuity at c, where a < c < b then

∫ (^) b a f^ (x)dx^ =^

∫ (^) c a f^ (x)dx^ +^

∫ (^) b c f^ (x)dx, provided both the limits on the right hand side exist and are finite.

EXAMPLE: The integral

a

1 xp^ dx^ is convergent if^ p >^ 1 and divergent if^ p^ ≤^ 1.