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What-are-the-conditions-for-using-L-hospital-rule-




Question Number 30429 by NECx last updated on 22/Feb/18
What are the conditions for using  L′hospital rule?
$${What}\:{are}\:{the}\:{conditions}\:{for}\:{using} \\ $$$${L}'{hospital}\:{rule}? \\ $$
Commented by prof Abdo imad last updated on 22/Feb/18
if f and g are n derivable at point x_0  with  f(x_0 )=f^, (x_0 )=...=f^((n−1)) (x_0 )=0 and f^((n)) (x_0 )≠0  also g(x_0 )=g^′ (x_0 )=...=g^((n−1)) (x_0 )=0 and g^((n)) (x_0 )≠0  so lim_(x→x_0 )   ((f(x))/(g(x)))= ((f^((n)) (x_0 ))/(g^((n)) (x_0 ))) .
$${if}\:{f}\:{and}\:{g}\:{are}\:{n}\:{derivable}\:{at}\:{point}\:{x}_{\mathrm{0}} \:{with} \\ $$$${f}\left({x}_{\mathrm{0}} \right)={f}^{,} \left({x}_{\mathrm{0}} \right)=…={f}^{\left({n}−\mathrm{1}\right)} \left({x}_{\mathrm{0}} \right)=\mathrm{0}\:{and}\:{f}^{\left({n}\right)} \left({x}_{\mathrm{0}} \right)\neq\mathrm{0} \\ $$$${also}\:{g}\left({x}_{\mathrm{0}} \right)={g}^{'} \left({x}_{\mathrm{0}} \right)=…={g}^{\left({n}−\mathrm{1}\right)} \left({x}_{\mathrm{0}} \right)=\mathrm{0}\:{and}\:{g}^{\left({n}\right)} \left({x}_{\mathrm{0}} \right)\neq\mathrm{0} \\ $$$${so}\:{lim}_{{x}\rightarrow{x}_{\mathrm{0}} } \:\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}=\:\frac{{f}^{\left({n}\right)} \left({x}_{\mathrm{0}} \right)}{{g}^{\left({n}\right)} \left({x}_{\mathrm{0}} \right)}\:. \\ $$

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