- #51

vela

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\begin{align*}

\left(x^3 +x^2 +\frac{x}{2}\right)e^{\frac{1}{x}} - \sqrt{x^6+1}

&= \left[\left(x^3 +x^2 +\frac{x}{2}\right)e^{\frac{1}{x}} - \sqrt{x^6+1}\right]\frac{\left(x^3 +x^2 +\frac{x}{2}\right)e^{\frac{1}{x}} + \sqrt{x^6+1}}{\left(x^3 +x^2 +\frac{x}{2}\right)e^{\frac{1}{x}} + \sqrt{x^6+1}} \\

&= \frac{\left(x^3 +x^2 +\frac{x}{2}\right)^2e^{\frac{2}{x}} - (x^6+1)}{\left(x^3 +x^2 +\frac{x}{2}\right)e^{\frac{1}{x}} + \sqrt{x^6+1}} \\

\end{align*} It's fairly straightforward to analyze the problem from here with the usual Calc 1 methods.