Saturday, 12 August 2017

real analysis - How to Find limlimitsxto0left(fractanxxright)frac1x2.

Can someone help me with this limit? I'm working on it for hours and cant figure it out.




limx0(tanxx)1x2



I started transforming to the form
limx0eln(tanxx)x2
and applied the l'Hopital rule (since indeterminated 00), getting:



limx0(2xsin2x2x2sin2x)



From here, I try continue with various forms of trigonometric substitutions, appling the l'Hopital rule again and again, but no luck for me. Can someone help me?

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