Draw and discuss the graph of the function `f(x) = e^(1//x)`

Explanation

We have `f(x) = e^(1//x)`
Clearly, the domain of function is `R-{0}`
Also `e^(1//x) gt 0`, hence the graph of the function lies above the x-axis.
`lim_(x to 0^(+))e^(1/x) = e^(infty)=infty` and `lim_(x to 0^(-))e^(1/x)=e^(-infty)=0`
`lim_(x to +-infty)e^(1/x) = e^(0)=1`
`f^(')(x)=e^(1/x)(-1/x^(2)) lt 0`. Hence, `f(x)` is a decreasing function in the domain.
From the above discussion, the graph of `y=f(x)` can be drawn as shown in the following figure.

As shown in the figure, `y=1` is an asymptote.