Saturday, 28 February 2015

indeterminate forms - Limit fracefracx22cos(x)x3sin(x) as xto0



We have to find the limit:




limx0ex22cos(x)x3sin(x)



I was stuck, but was able to find the limit using series expansion as 14.



How can we calculate the limit with standard limits like



limx0ex1x=1limx0sin(x)x=1



etc.




Also I didn't try L'hospital as that would be too complicated.


Answer



Using Taylor polynomials, you have
1x22+x48+O(x6)(1x22+x424+O(x6))x3sinx=x412+O(x6)x3sinx112.


You cannot expect to use limits as simple as those in your question, because this limit depends on the terms of degree two in the expansion, while the two limits you quote depend on the terms of degree one.


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