If $\underset{x→∞}{\lim}(1+\frac{λ}{x}+\frac{μ}{x^2})^{2x}=e^2$, then

λ = 1, μ = any real constant

$\underset{x→∞}{\lim}\begin{pmatrix}1+\frac{λx+μ}{x^2}\end{pmatrix}^{2x}=\underset{x→∞}{\lim}\begin{pmatrix}1+\frac{1}{\frac{x^2}{λx+μ}}\end{pmatrix}^{\frac{x^2}{λx+μ}.\frac{2(λx+μ)}{x}}$

$=\begin{bmatrix}\underset{x→∞}{\lim}\begin{pmatrix}1+\frac{1}{\frac{x^2}{λx+μ}}\end{pmatrix}^{\frac{x^2}{λx+μ}}\end{bmatrix}^{\underset{x→∞}{\lim}\frac{2(λx+μ)}{x}}=e^{\underset{x→∞}{\lim}\begin{pmatrix}2λ+\frac{2μ}{x}\end{pmatrix}}=e^{2λ}$

$∴e^{2λ}=e^2$  $∴λ=1$