Sunday, 18 August 2013

limits - Find limxto0frac1x3left(left(frac2+cosx3right)x1right) without L'Hopital's rule




Find the following limit

limx01x3((2+cosx3)x1)


without using L'Hopital's rule.




I tried to solve this using fundamental limits such as limx0(1+x)1x=e,limx0xx=1 and equivalent infinitesimals at x0 such as xsinx,ax1+xlna,(1+x)a1+ax. This is what I did so far:
limx01x3((2+cosx3)x1)=limx0(3(1cosx)3)x1x3=limx0(312x23)x1x3=limx0(116x2)x1x3


I used fact that xsinx at x0. After that, I tried to simplify (116x2)x. Problem is because this is not indeterminate, so I cannot use infinitesimals here. Can this be solved algebraically using fundamental limits, or I need different approach?


Answer



You are certainly on the right track (although you've used an approximation that needs justification). Now write  (1x2/6)x=exp(xln(1x2/6)). More approximations: ln(1h)h and eh1+h for h small. What happens if each is an =? That will tell you what the limit is. Now you need to make sure these approximations really work.


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