## Question

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

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

## Answers

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)

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