calculus and analysis – How to use Mathematica to evaluate the limit of Exponential Functions?

Clear("Global`*")

What you wrote is

(E^-2 x + 3 x)/x // Simplify

(* 3 + 1/E^2 *)

Presumably you meant to write

expr = (E^(-2 x) + 3 x)/x;

First, check that l’Hopital’s rule is applicable

Limit(#, x -> Infinity) & /@ {Numerator(expr), Denominator(expr)}

(* {∞, ∞} *)

The derivatives are

D({Numerator(expr), Denominator(expr)}, x)

(* {3 - 2 E^(-2 x), 1} *)

The limits of the derivatives are

Limit(%, x -> Infinity)

(* {3, 1} *)

The ratio of the limits is the limit of expr

Divide @@ %

(* 3 *)

Verifying,

Limit(expr, x -> Infinity)

(* 3 *)