```
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 *)
```