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what is derivation of L hospital rule

7 years ago

SAGAR SINGH - IIT DELHI
879 Points

Dear vikash,

Let lim stand for the limit , , , , or , and suppose that lim and lim are both zero or are both . If

has a finite value or if the limit is , then

Historically, this result first appeared in l'Hospital's 1696 treatise, which was the first textbook on differential calculas.

All the best.

Sagar Singh

B.Tech IIT Delhi

Comes as a direct consequence of the Mean Value theorem. Useful for

computing limits of the formf(x)

g(x)wheref(x),g(x) 0. Connected to

limxa

f?(x)

g?(x). Assumef (a) =g(a) = 0, and you want to compute limxa

f(x)

g(x).

Use the mean value theorem to try to approximate the function. Asx ap- proachesa,f (x) should be approximatelyf (a) +f?(y)(x a). Using the mean value theorem, we know such ay exists, sof (x) =f (a) +f?(y)(x a) for somea < y < x. Likewise,g(x) =g(a) +g?(z)(x a) for somea < z < x. So

f(x)

g(x)=f?(y)(x a)

g?(z)(x a)=f?(y)

g?(z)f?(x)

f?(y)

sincey andz approachx asx approachesa. Keep this idea in mind for the

proof.

2 Proof

Theorem (generalized mean value theorem): letf, g be continuous on [a, b],

and di?erentiable on (a, b). Thenx (a, b) such thatf?(x)· (g(b)g(a)) =

g?(x)(f(b) f(a)).

f?(x)

g?(x)=f(b) f(a)

g(b) g(a)

7 years ago
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