Find all asymptotes of y=x e^(1/x)

Find all asymptotes of $y=x\,e^{1/x}$.

Step 1: Behaviour at $x=0$

As $x\to0^+$, $1/x\to+\infty$ so $e^{1/x}\to+\infty$ and $xe^{1/x}\to+\infty$: $x=0$ is a vertical asymptote from the right. As $x\to0^-$, $1/x\to-\infty$ so $e^{1/x}\to0$ and $xe^{1/x}\to0$: no asymptote on that side.

Step 2: Oblique asymptote — slope

\[ k = \lim_{x\to\pm\infty} e^{1/x} = 1. \tag{1} \]

Step 3: Oblique asymptote — intercept

Substituting $t=1/x\to0$: \[b = \lim_{x\to\pm\infty} x\bigl(e^{1/x}-1\bigr) = \lim_{t\to0}\frac{e^t-1}{t} = 1. \tag{2}\]

Step 4: Verification

\[x=1000\text{: } y-(x+1)\approx0.0005\to0;\qquad x=0.001\text{: }y\approx2.688\times10^{434}\ (\to+\infty)\ \checkmark\]