Question

What are the critical points of `f(x) =1/(xe^x)-e^(2x)/x`?

Answer 1

`"f"(x)=(1-e^(3x))/(xe^x)`

By using the series for the exponential function,

`"f"(x)=(1-(1+3x+(3x)^2/2!+...))/(x+x^2+x^3/2!+...)`

`"f"(x)=(3x+(3x^2)/2!+...))/(x+x^2+x^3/2!+...)`

`"f"(x)=(3+9x/2!+...)/(1+x+...)`

Then as `x \to 0`, `"f"(x) \to 3`.