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Ratio limits of fibonacci-like series

# Definition
\$F_{n}\left(0\right)=0\$
\$F_{n}\left(1\right)=1\$
\$F_{n}\left(x\right)=n\cdot F_{n}\left(x-1\right)+F_{n}\left(x-2\right)\$
For example:
\$F_{1}=\left[0,1,1,2,3,5,8,13,21,34,55,89...\right]\$
\$F_{2}=\left[0,1,2,5,12,29,70,169,408,985...\right]\$
\$F_{3}=\left[0,1,3,10,33,109,360,1189,3927...\right]\$
# Challenge
Given n, find the limit of the ratio between consecutive terms, correct to at least the first 5 decimal places.
# Test Cases
```
1 -> 1.61803...
2 -> 2.41421...
3 -> 3.30277...
4 -> 4.23606...
```
# Scoring
~~This is code-golf. Shortest answer in each language wins.~~

# Definition
\$F_{n}\left(0\right)=0\$
\$F_{n}\left(1\right)=1\$
\$F_{n}\left(x\right)=n\cdot F_{n}\left(x-1\right)+F_{n}\left(x-2\right)\$
For example:
\$F_{1}=\left[0,1,1,2,3,5,8,13,21,34,55,89...\right]\$
\$F_{2}=\left[0,1,2,5,12,29,70,169,408,985...\right]\$
\$F_{3}=\left[0,1,3,10,33,109,360,1189,3927...\right]\$
# Challenge
Given n, find the limit of the ratio between consecutive terms, correct to at least the first 5 decimal places.
# Test Cases
```
1 -> 1.61803...
2 -> 2.41421...
3 -> 3.30277...
4 -> 4.23606...
```
# Scoring
This is <a class="badge is-tag">code-golf</a>. Shortest answer in each language wins.

code-golf sequence