Question

The problem is attached

Answer 1

$$\begin{align*}(xe^{9x}+5x)^{1/3x}&=\exp\left(\frac1{3x}\log(xe^{9x}+5x)\right)\\&=\exp\left(\frac1{3x}(\log x+\log (e^{9x}+5))\right)\\&\approx\exp\left(\frac1{3x}\log(e^{9x})\right)\\&=e^3\end{align*}$$

Answer 2

since in the long run \(\log(e^{9x}+5)\sim\log(e^{9x})=9x\) and we know \(\dfrac1{3x}\log x\to 0\)

Answer 3

Here is the solution generated by http://saab.org