Calculus
Lesson
46
L'Hopitals
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There are 7 indeterminate FORMS:
These are indeterminate when BOTH TERMS approach the values.
Note that 
!!!!! But 
since we don't know how
to look at it.
Note that the FORMS 
are
all identical. If we look at
the term 
, that we can just
rewrite it as 
or
.
Whenever you take the limit of a form that looks like 
, we use
L'Hopital's Rule to solve for the limit.
Recall from last year: to find some limits, we simply cancel some terms:
However, sometimes we can't cancel anything!
looks like 
when 
, but there's nothing to
cancel!
Use a calculator to examine Y1(-0.2), Y1(-0.1), Y1(-0.001), Y1(0.2), Y1(0.1), Y1(0.001).
Before we say what L'Hopital's Rule is, let's derive it! Let's say that we have
two functions, both of which = 0 when x = a. Then when we try to look at
, we get 
, which is indeterminate.
BUT, recall that as x gets
closer and closer to a, which is the same as saying that we are "zooming in" at
x=a, that ANY function looks like a line. The equation of that line, using
point-slope form (and given that the point (a,0) is a point we know and we
know the slope is 
),
is
. The same can be
said for 
.
SO, ONLY WHEN WE GET CLOSE TO x=a, we find:
. Now, we can cross out
the 
s
since x NEVER equals a (it only gets really close). What we're left with is:
This is L'Hopital's Rule!!!!
ONLY works if 
is
undefined! It DOES NOT work otherwise!
Both functions must also be continuous, except for perhaps at x=a.
Finishing our example, then, 
Other examples: 
Okay, so what about other forms? If the form you have is 
, then
get everything over a common denominator or factor, then use L'Hopital's.
Examples: 
If you have one of the exponential forms, 
, set the limit
equal to y, take the log of both sides, and THEN use L'Hopital's:
Examples: 
