What is #lim_(xrarroo) (e^xsin(1/x))/x #?

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Answer 1 · 2015-11-06T15:57:43

The limit does not exist.

As #x->oo#, #1/x->0^+#, #sin(1/x)->1/x#.

#frac{e^xsin(1/x)}{x}~~frac{e^x(1/x)}{x}=frac{e^x}{x^2}#

Since exponential growth (numerator) is faster than polynomial growth (denominator), the limit does not exist.
(http://math.stackexchange.com/questions/872848/proving-exponential-is-growing-faster-than-polynomial)