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CHAPTER 2
2.4 Continuity
Indeterminate
Forms and
L’Hospital’s Rule
L’Hospital’s Rule Suppose f and g are
differentiable and g’(x)  0 near a.
Suppose that
lim x a f(x) = 0 and lim x a g(x) = 0, or
lim x a f(x) = oo and lim x a g(x) = oo
Then lim x a f(x) / g(x) = lim x a f’(x)/g’(x)
if the limit on the right side exists.
Example Find
lim
x  tan x /x.
CHAPTER 2
2.4 Continuity
x
Example Find
lim
x oo ln (1 + e ) / 5x.
CHAPTER 2
2.4 Continuity
Indeterminate Products
If lim x a f(x) = 0 and lim x a g(x) = oo,
then it’s not clear what the value of
lim x a f(x)g(x). We can find the value of
this limit by writing the product fg as a
quotient :
fg = f / (1 / g)
or
fg = g / (1 / f ).
Example Find lim x oo x ex .
Indeterminate Differences
If lim x a f(x) = oo and lim x a g(x) = oo,
then the limit
lim x a [ f(x) - g(x) ]
is called an indeterminate form of type
oo - oo .
Example Find lim x 0 ( csc x – cot x ).
Indeterminate Powers
Several indeterminate forms arise from
lim x a [ f(x) ]g(x) .
1. lim x a f(x) = 0 and lim x a g(x) = 0
(type 00).
2. lim x a f(x) = oo and lim x a g(x) = oo
(type oo0 ).
3. lim x a f(x) = 1 and lim x a g(x) =  oo
(type 1oo ).
Example Calculate
lim x oo ( 1 + ( a / x ))bx.