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If there is a point x=a, where f(x) and g(x) intersect, then you can take the limit as x approaches "a" of f(x) and g(x) separately. The ratio of the 2 instantaneous rate of changes at x=a will be the limit as the orginal function f(x) / g(x) approaches "a".
(1/x)
lim X
x->∞
Indeterminate form is a ratio that you get when you use substitution to solve the limit.
When you substitute, you get a ratio that does not give you a real answer.
This is where L' Hopital's rule comes in handy.
IF
Lim f(x)=Lim g(x)=0 or (+/-)∞
X->C
Work Book Page 30, #12
exists and g'(x) ≠ 0
and Lim f’(x)
___
cos(2x)-1 , x≠0
{
f(x)=
m , x=0
g'(x)
X->C
Then Lim f(x) = Lim f'(x)
___
g'(x)
g(x)
X->C
1
____
(+/-) ∞ - (+/-) ∞
If we get this as our ratio when subbing, we would use algebra to get it into the proper indet form.
1∞
If we get this, then we would take the natural log which would bring down the exponent. Then we use algebra to get intot the proper indet form if its not already in it.
-
( )
ln(x)
(x-1)
X->1