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

Nov 6, 2015

The limit does not exist.

#### Explanation:

$\sin \left(\theta\right) \approx \theta$ at small $\theta$ (in radians).
(http://www.ies-math.com/math/java/calc/LimSinX/LimSinX.html)

As $x \to \infty$, $\frac{1}{x} \to {0}^{+}$, $\sin \left(\frac{1}{x}\right) \to \frac{1}{x}$.

$\frac{{e}^{x} \sin \left(\frac{1}{x}\right)}{x} \approx \frac{{e}^{x} \left(\frac{1}{x}\right)}{x} = \frac{{e}^{x}}{{x}^{2}}$

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)