L'Hospital's Rule

Author

Eric W. Weisstein
October 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/LHospitalsRule.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LHospitalsRule.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Oscillates

LHospitalStep[f_,u_]:=Divide@@D[Through[{Numerator,Denominator}[f]],u]

NestList[LHospitalStep[#,u]&,(u/(Sqrt[(u^2)+1])),5]

Limit[(u/(Sqrt[1+(u^2)])),u->∞]

Plot[(u/(Sqrt[(u^2)+1])),{u,-1,1},PlotStyle->Red,ImageSize->200,AxesLabel->{u,(u/(Sqrt[(u^2)+1]))}]

Counterexample

<<Utilities`Typesetting`

Boas

λ[x_]:=Sin[x]
f[x_]:=Integrate[Derivative[1][λ][t]^2,{t,0,x}]
ϕ[x_]:=Exp[x]
g[x_]:=f[x]ϕ[λ[x]]

{f[x],g[x]}//Simplify

Plot[{f[x],g[x]},{x,-10,10},PlotRange->All,PlotStyle->{Red,Blue},
AxesLabel->TraditionalForm/@{x,{StyleForm[f[x]//Simplify,FontColor->Red],StyleForm[g[x]//Simplify,FontColor->Blue]}}]

Plot[((f[x])/(g[x])),{x,-1,100}]

Limit[{f[x],g[x]},x->∞]

Limit[Divide@@{f[x],g[x]},x->∞]

Limit[Divide@@{f'[x],g'[x]},x->∞]

Rickert

f[x_]:=x Sin[1/x^4]Exp[-1/x^2]
g[x_]:=Exp[-1/x^2]

Plot[{f[x],g[x]},{x,-2,2},PlotRange->All,PlotStyle->{Red,Blue},
AxesLabel->TraditionalForm/@{x,{StyleForm[f[x],FontColor->Red],StyleForm[g[x],FontColor->Blue]}/.Power[x_,n_Integer?Negative]:>InlineFraction[1,x^(-n)]}]

Limit[Divide@@{f[x],g[x]},x->0]

Limit[Divide@@{f'[x],g'[x]},x->0]