Saturday, 24 September 2016

calculus - $ lim_{x to infty} frac {e^{x/2}}{2^x} $



How do i know what is the limit :



$$ \lim_{x \to \infty} \frac {e^{x/2}}{2^x} $$



Because with l'hopital it is imposible to say, as the nth derivatives of both $ {e^{x/2}} $and $ {2^x}$ are always infinity...
I could use a hint that helps me think this through


Answer






$$\large\frac{e^{x/2}}{2^x}=\frac{e^{x/2}}{e^{x\ln(2)}}=e^{x\left[\frac12-\ln(2)\right]}\to\ ...$$


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