Saturday, December 9, 2017

calculus - Verify limlimitsnrightarrowinftyintinfty0!enxsin(ex),mathrmdx=0.



How to verify limn0enxsin(ex)dx=0?



My idea is to use the dominant convergence theorem with fn(x):=enxsin(ex) and f(x):=limnfn(x).




limn0enxsin(ex)dx=0limnenxsin(ex)dx=0limn0dx=0



Can I use this here?


Answer




How to verify limn0enxsin(ex)dx=0




You may just observe that

|0enxsin(ex)dx|0|enx|dx=1n,(n>0), and then let n+.


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