Question

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

Answer 1

The limit does not exist.

Explanation:

`sin(theta)~~theta` at small `theta` (in radians).
(http://www.ies-math.com/math/java/calc/LimSinX/LimSinX.html)

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)