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Explanation of L'Hopital's Rule

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Date: 02/06/2001 at 23:34:26
From: Samuel Ward
Subject: L'Hospital's Rule

In certain cases, L'Hopital's Rule connects the limit of a quotient
(f/g) to the limit of the quotient of the derivatives (f'/g'). This
is true when f and g go to 0 or infinity at the point where the limit
is taken.

I understand how to use this rule, and I somewhat understand the
proof, but I still do not understand why this happens. Can you help?
Please also try to describe explicitly how to think of the roles of
the limit, the derivative, and the quotient.
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Date: 02/07/2001 at 20:55:05
From: Doctor Fenton
Subject: Re: L'Hospital's Rule

Hi Samuel,

Thanks for writing to Dr. Math. You've posed a very good question. One
way you can think of this is to use the idea of derivative: a function
f(x) is differentiable at x=a if f(x) is very close to its tangent
line y = f'(a)*(x-a) + f(a) near x = a. Specifically,

f(x) = f(a) + f'(a)*(x-a) + E1(x)

where E1(x) is an error term which goes to 0 as x goes to a. In fact,
E1(x) must approach 0 so fast that

E1(x)
lim  ----- = 0
x->a  x-a

because

E1(x)   f(x)-f(a)
----- = --------- - f'(a)
x-a       x-a

and we know from the definition of derivative that this quantity has
the limit 0 at a.

Similarly, if g is differentiable at x = a,

g(x) = g(a) + g'(a)*(x-a) + E2(x)

where E2(x) is another error term which goes to 0 as x->a. If you're
computing the limit of f(x)/g(x) as x->a and if g(a) is not equal to
0, then as x->a, the numerator becomes indistinguishable from f(a)
and the denominator from g(a), so the limit is

lim    f(x)   f(a)
x->a   ---- = ----
g(x)   g(a)

If both f(a) and g(a) are 0, then we must use the tangent
approximations to say that

f(x)   f(a) + f'(a)*(x-a) + E1(x)
---- = --------------------------
g(x)   g(a) + g'(a)*(x-a) + E2(x)

f'(a)*(x-a) + E1(x)
= ---------------------
g'(a)*(x-a) + E2(x)

f'(a) + [E1(x)/(x-a)]
= ---------------------
g'(a) + [E2(x)/(x-a)]

and we have seen that the second term becomes negligible as x->a.

In other words, when both function values approach 0 as x->a, the
ratio of the function values just reduces to the ratio of the slopes
of the tangents, because both functions are very close to their
tangent lines.

Does this clarify the situation? If you still have questions, please
write again.

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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