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L'Hôpital's Rule

 

Kevin Aylward B.Sc.

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Overview

This paper provides a derivation of L'Hôpital's Rule.

L'Hôpital's Rule

Consider:

f ′ (x)= lim Δx→0 f(x+Δx)−f(x) Δx g ′ (x)= lim Δx→0 g(x+Δx)−g(x) Δx lim Δx→0 f ′ (x) g ′ (x) = f(x+Δx)−f(x) Δx g(x+Δx)−g(x) Δx = f(x+Δx)−f(x) g(x+Δx)−g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@A9D0@

if f(x)=g(x)=0 then MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabMgacaqGMbGaaeiiaiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGNbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimaiaabccacaqG0bGaaeiAaiaabwgacaqGUbaaaa@4608@

lim Δx→0 f ′ (x) g ′ (x) = f(x+Δx) g(x+Δx) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaciiBaiaacMgacaGGTbaaleaacqqHuoarcaWG4bGaeyOKH4QaaGimaaqabaGcdaWcaaqaaiqadAgagaqbaiaacIcacaWG4bGaaiykaaqaaiqadEgagaqbaiaacIcacaWG4bGaaiykaaaacqGH9aqpdaWcaaqaaiaadAgacaGGOaGaamiEaiabgUcaRiabfs5aejaadIhacaGGPaaabaGaam4zaiaacIcacaWG4bGaey4kaSIaeuiLdqKaamiEaiaacMcaaaaaaa@52D4@

Taking the limit Δx→0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaadIhacqGHsgIRcaaIWaaaaa@3AF3@

f ′ (x) g ′ (x) = f(x) g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGabmOzayaafaGaaiikaiaadIhacaGGPaaabaGabm4zayaafaGaaiikaiaadIhacaGGPaaaaiabg2da9maalaaabaGaamOzaiaacIcacaWG4bGaaiykaaqaaiaadEgacaGGOaGaamiEaiaacMcaaaaaaa@442D@

For the case of if f(x)=g(x)=∞ then MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabMgacaqGMbGaaeiiaiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGNbGaaiikaiaadIhacaGGPaGaeyypa0JaeyOhIuQaaeiiaiaabshacaqGObGaaeyzaiaab6gaaaa@46BF@ , as above

lim Δx→0 f ′ (x) g ′ (x) = f(x+Δx)−f(x) g(x+Δx)−g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaciiBaiaacMgacaGGTbaaleaacqqHuoarcaWG4bGaeyOKH4QaaGimaaqabaGcdaWcaaqaaiqadAgagaqbaiaacIcacaWG4bGaaiykaaqaaiqadEgagaqbaiaacIcacaWG4bGaaiykaaaacqGH9aqpdaWcaaqaaiaadAgacaGGOaGaamiEaiabgUcaRiabfs5aejaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWG4bGaaiykaaqaaiaadEgacaGGOaGaamiEaiabgUcaRiabfs5aejaadIhacaGGPaGaeyOeI0Iaam4zaiaacIcacaWG4bGaaiykaaaaaaa@5B31@

Dividing out by g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacaGGOaGaamiEaiaacMcaaaa@392B@

lim Δx→0 f ′ (x) g ′ (x) = f(x+Δx) g(x) − f(x) g(x) g(x+Δx) g(x) − g(x) g(x) lim Δx→0 f ′ (x) g ′ (x) = f(x+Δx) g(x) − f(x) g(x) g(x+Δx) g(x) −1 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@9538@

As , Δx≠0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaadIhacqGHGjsUcaaIWaaaaa@3ACD@  and g(x)→∞ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacaGGOaGaamiEaiaacMcacqGHsgIRcqGHEisPaaa@3C89@

the first terms of the denominator and numerator are 0

on taking the g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacaGGOaGaamiEaiaacMcaaaa@392B@  limit first, thus

lim Δx→0 f ′ (x) g ′ (x) = 0− f(x) g(x) 0−1 = f(x) g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaciiBaiaacMgacaGGTbaaleaacqqHuoarcaWG4bGaeyOKH4QaaGimaaqabaGcdaWcaaqaaiqadAgagaqbaiaacIcacaWG4bGaaiykaaqaaiqadEgagaqbaiaacIcacaWG4bGaaiykaaaacqGH9aqpdaWcaaqaaiaaicdacqGHsisldaWcaaqaaiaadAgacaGGOaGaamiEaiaacMcaaeaacaWGNbGaaiikaiaadIhacaGGPaaaaaqaaiaaicdacqGHsislcaaIXaaaaiabg2da9maalaaabaGaamOzaiaacIcacaWG4bGaaiykaaqaaiaadEgacaGGOaGaamiEaiaacMcaaaaaaa@57FC@

Taking the limit Δx→0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaadIhacqGHsgIRcaaIWaaaaa@3AF3@

f ′ (x) g ′ (x) = f(x) g(x) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGabmOzayaafaGaaiikaiaadIhacaGGPaaabaGabm4zayaafaGaaiikaiaadIhacaGGPaaaaiabg2da9maalaaabaGaamOzaiaacIcacaWG4bGaaiykaaqaaiaadEgacaGGOaGaamiEaiaacMcaaaaaaa@442D@

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