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DEBABRATA NAG

To prove L'Hopital's Rule (sometimes spelled L'Hospital's Rule), we
use the Taylor expansion:

f(a+h) = f(a) + hf'(a) + terms in h^2 and higher
g(a+h) = g(a) + hg'(a) + terms in h^2 and higher

So:Neglecting higher powers......

...........f(a+h) ...................f(a)+h*f'(a)
Lt........ ------....... ->.......... ------------
h->0 ...g(a+h) ..................g(a)+h*g'(a)

so with f(a) = g(a) = 0 we get:

................f(a+h) ....................h*f'(a)..................... f'(a)
Lt............. -------......... ->........ ------- ........->......... ------
h->0 ........g(a+h) ...................h*g'(a) .....................g'(a)

We can use l'Hopital's also if f'(a) -> infinity and g'(a) -> infinity:

f(a)........................ infinity .........................1/g(a)........................ 0
----......... ->............ -------- ...........so.......... --------........... ->......... ---
g(a)....................... infinity .........................1/f(a)......................... 0

and applying l'Hopital's to this latter expression, we get:

f(a) .....................-g'(a)/[g(a)]^2 ........................g'(a)*[f(a)]^2
------........ ->........ ----------------......... ->............ ----------------
g(a) ....................-f'(a)/[f(a)]^2.......................... f'(a)*[g(a)]^2

and cross-multiplying:

f'(a)....... f(a)
------- -> ------
g'(a)..... g(a)

Therefore whether we have 0/0 or infinity/infinity we can use
l'Hopital's rule.

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