Finding asymptotes of exponential function and one-sided limit Find the asymptotes of
$$
\lim_{x \to \infty}x\cdot\exp\left(\dfrac{2}{x}\right)+1.
$$
How is it done?

A related problem. We will use the Taylor series of the function $e^t$ at the point $t=0$,
$$ e^t = 1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots .$$
$$ x\,e^{2/x}+1 = x ( 1+\frac{2}{x}+ \frac{1}{2!}\frac{2^2}{x^2}+\dots )+1=x+3+\frac{2^2}{2!}\frac{1}{x}+\frac{2^3}{3!}\frac{1}{x^2}+\dots$$
$$ = x+3+O(1/x).$$
Now, you can see when $x$ goes to infinity, then you have
$$ x\,e^{2/x}+1 \sim x+3 $$
Here is the plot of $x\,e^{2/x}+1$ and the Oblique asymptote $x+3$ 

