<s>[INST] A tricky question on sequences of functions. Find
$$\lim_{n \to \infty}\left[
{1 \over x + 1} + {2x \over \left(x + 1\right)\left(x + 2\right)}
+{3x^{2} \over \left(x + 1\right)\left(x + 2\right)\left(x + 3\right)}
+ \cdots
+ {nx^{n-1} \over
\left(x + 1\right)\left(x + 2\right)\ldots\left(x + n\right)}\right]$$
 [/INST] HINT:
For integer $r\ge1$,
$$\frac{rx^{r-1}}{(x+1)(x+2)\cdots(x+r-1)(x+r)}+\frac{x^r}{(x+1)(x+2)\cdots(x+r-1)(x+r)}=\frac{x^{r-1}}{(x+1)(x+2)\cdots(x+r-1)}$$
$$\iff  \frac{rx^{r-1}}{(x+1)(x+2)\cdots(x+r-1)(x+r)}$$
$$=\frac{x^{r-1}}{(x+1)(x+2)\cdots(x+r-1)}-\frac{x^r}{(x+1)(x+2)\cdots(x+r-1)(x+r)} $$
Set $r=1,2,3,c\dots, n-1,n$ to add to recognize the Telescoping Series
Finally set $n\to\infty$
</s>