Infinite product

Relevant wiki: Riemann Sums

$\begin{aligned} P & = \lim_{n \to \infty} \prod_{r=1}^{2n} \left(2+\frac rn \right) \\ & = \lim_{n \to \infty} \exp \left(\sum_{r=1}^{2n} \frac 1n \ln \left(2+\frac rn \right) \right) \\ & = \exp \left( \lim_{n \to \infty} \sum_{r=1}^{2n} \frac 1n \ln \left(2+\frac rn \right) \right) & \small \color{#3D99F6} \text{Riemann's sum: } \lim_{n \to \infty} \sum_{r=a}^b f \left(\frac rn \right) = \int_{\lim_{n \to \infty} a/n}^{\lim_{n \to \infty} b/n} f(x) \ dx \\ & = \exp \left(\int_0^2 \ln (2 + x) \ dx \right) & \small \color{#3D99F6} \text{Let } u = 2+x, \ du = dx \\ & = \exp \left(\int_2^4 \ln u \ du \right) & \small \color{#3D99F6} \text{By integration by parts.} \\ & = \exp \left( u \ln u \bigg|_2^4 - \int_2^4 du \right) \\ & = \exp \left( u \ln u - u \bigg|_2^4 \right) \\ & = \exp \left( 4 \ln 4 - 4 - 2 \ln 2 + 2 \right) \\ & = \exp \left( 6 \ln 2 - 2 \right) \\ & = \frac {2^6}{e^2} \end{aligned}$

$\implies \dfrac b{2+c} = \dfrac 6{2+2} = \boxed{1.5}$

Let the limit equal L. Take log on both sides. The right hand side gets converted into a Riemann Sum. This yields the values of b and c quite easily