Why is $\lim_{x \to \infty}\left( \frac{x-1}{e^{1/x}} - x \right) = -2$? Please, I need the method of doing the following limit:
$$
 \lim_{x \to \infty}\left( \frac{x-1}{e^{1/x}} - x \right) = -2.
$$

Set $1/x=u\implies u\to0$
$$\lim_{x \to \infty}\left( \frac{x-1}{e^{1/x}} - x \right) =\lim_{u\to0}\left(\dfrac{1-u}{ue^u}-\dfrac1u\right)$$
$$=\left(\lim_{u\to0}\dfrac1{e^u}\right)\cdot\left(-1-\lim_{u\to0}\dfrac{e^u-1}u\right)$$
Can you take it from here?
