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Typology: Cheat Sheet
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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.
lim x→a
f (x) g(x)
= lim x→a
f ′(x) g′(x)
lim x→a
f (x) g(x)
= lim x→a
f ′(x) g′(x)
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.
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.
NOTE THAT THESE ARE THE ONLY TYPES OF INDETERMINATE FORMS.
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 −∞ 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.
∞ f^ (x)dx^ =^
∫ (^) a −∞ f^ (x)dx^ +^
a f^ (x)dx^ for any real number^ a, provided both the integrals are convergent.
∫ (^) b a f^ (x)dx^ = limt→b−
∫ (^) t a f^ (x)dx.
∫ (^) b a f^ (x)dx^ = limt→a+
∫ (^) b t f^ (x)dx.
∫ (^) 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.