L'hopital's rule

1. We can already take many limits using techniques and reasoning independent of
calculus. The fundamental concepts of calculus lead to more techniques related to
limits. One such technique used for taking limits is called L’Hôpital’s Rule, named
after the French mathematician who made it famous, even though it is a natural
extension of differential calculus.

L’Hôpital’s Rule states that: 
xc
lim
f(x)
g(x)

f'(c)
g'(c)
if f(c) = 0, g(c) = 0 and g’(c) does not equal zero.

It is important to remember that a limit is something that is approached rather than
reached, and so taking the limits of the numerator and denominator of the
expression separately and then taking the quotient of the limits tells us nothing if
both limits are zero. What we care about is not the ratio of f(x) to g(x) at c, but the
ratio of f(x) to g(x) as x approaches c. So what this means is to take the ratio of f(x)
to g(x) infinitely close to c, but not at c. If f(c) = 0, g(c) = 0, both functions are
differentiable at c and g’(c) does not equal zero, then 
xc
lim
f(x)
g(x)

f'(c)
g'(c)
, because since
f(ch)f'(c)hf(c) and 
g(ch)g'(c)hg(c)
if h is an infinitely small change in x known as a differential, and because f(c) = 0 and
g(c) = 0, 
f(ch)f'(c)h and
g(ch)g'(c)h , 
xc
lim
f(x)
g(x)

f(ch)
g(ch)

f'(c)h
g'(c)h

f'(c)
g'(c)
. If h represents an infinitely small change in x,
then
f(ch)f'(c)hf(c) and
g(ch)g'(c)hg(c) are natural rearrangements
of the definitions of
f'(c) and
g'(c) . This technique works when f(c) = g(c) = 0 and 
f'(c)
and
g'(c) exist. L’Hôpital’s Rule should also work if 
xc
lim
f(x)
xc
lim
g(x)0
since taking the quotient 
xc
lim
f(x)
xc
lim
g(x) does not necessarily equate to dividing f(c) by
g(c). For example, if there is a hole or jump in the graph of at least one of the
functions at x = c, at least one of the functions is not continuous at x = c and thus its
limit at x approaches c does not equal its value at x = c. Also, the function that is not
continuous at x = c is also not differentiable at x = c, so
f'(c)
g'(c) does not exist.
However, 
xc
lim
f'(x)
g'(x) does exist if 
xc
lim
f'(x) exists and 
xc
lim
g'(x) exists, which
requires that the one-sided limits of each individual derivative function are equal.

The form of L’Hôpital’s Rule with fewer requirements states that

2. 
xc
lim
f(x)
g(x)

xc
lim
f'(x)
g'(x) .

This more generalized form of L’Hôpital’s Rule requires that 
xc
lim
f(x)
xc
lim
g(x)0
, 
xc
lim
f'(x) exists and 
xc
lim
g'(x) exists.