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The third Painlev\'e equation in its generic form, often referred to asPainlev\'e-III($D_6$), is given by\[\dfrac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left(\dfrac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x}\dfrac{\mathrm{d}u}{\mathrm{d}x}+\dfrac{\alpha u^2 +\beta}{x}+4u^3-\frac{4}{u}, \quad \alpha,\beta \in \mathbb{C}.\] Starting froma generic initial solution $u_0(x)$ corresponding to parameters $\alpha,\beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicitB\"acklund transformation to generate a family of solutions$(u_n(x),\alpha+4n,\beta+4n)$ indexed by $n\in \mathbb{N}$. We study the large$n$ behavior of the solutions $(u_n(x),\alpha+4n,\beta+4n)$ under the scaling$x = z/n$ in two different ways: (a) analyzing the convergence properties ofseries solutions to the equation, and (b) using a Riemann-Hilbertrepresentation of the solution $u_n(z/n)$. Our main result is a proof that thelimit of solutions $u_n(z/n)$ exists and is given by a solution of thedegenerate Painlev\'e-III equation, known as Painlev\'e-III($D_8$),\[\dfrac{\mathrm{d}^2U}{\mathrm{d}z^2}=\dfrac{1}{U}\left(\dfrac{\mathrm{d}U}{\mathrm{d}z}\right)^2-\dfrac{1}{z}\dfrac{\mathrm{d}U}{\mathrm{d}z}+\dfrac{4U^2 + 4}{z}.\] A notable applicationof our result is to rational solutions of Painlev\'e-III($D_6$), which areconstructed using the seed solution $(1,4m,-4m)$ where $m\in \mathbb{C}\setminus (\mathbb{Z} + \frac{1}{2})$ and can be written as a particular ratioof Umemura polynomials. We identify the limiting solution in terms of both itsinitial condition at $z = 0$ when it is well-defined, and by its monodromy datain general case. Furthermore, as a consequence of our analysis, we deduce theasymptotic behavior of generic solutions of Painlev\'e-III, both $D_6$ and$D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemurapolynomials in a neighborhood of $z = 0$.

Painlev\'e-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions