I mean there exist $\lambda>0, x\in \mathbb R$ s.t. the inequality $$ \frac{3^{\lambda } e^x+e^{3 x}+1}{2^{\lambda } e^{2 x}}<\frac{1}{10^{100}}$$ is valid.

Here are my unsuccessful attempts.

```
Resolve[Exists[\[Lambda], \[Lambda] > 0, Exists[x, (1 + 3^\[Lambda]*E^x + E^(3 x))/2^\[Lambda]/E^(2 x) < 
10^-100]], Reals]

```

returns the input. The same issue with

```
Minimize[{(1 + 3^\[Lambda]*E^x + E^(3 x))/2^\[Lambda]/E^(2 x), \[Lambda] > 0}, {\[Lambda], x}]

```

and

```
FindInstance[(1 + 3^\[Lambda]*E^x + E^(3 x))/2^\[Lambda]/E^(2 x) < 
1/10^100 && \[Lambda] > 0, {\[Lambda], x}, Reals]

```

The result of the command

```
NMinimize[{(1 + 3^\[Lambda]*E^x + E^(3 x))/2^\[Lambda]/E^(2 x), \[Lambda] > 0}, {\[Lambda], x}]

```

> 
> `{0., {\[Lambda] -> 425.496, x -> 236.227}}`
> 
> 
> 

is not any proof because `0.` may be greater than `0`.