$\\ \lim_{x\rightarrow \infty } \dfrac {x(1+a/x)}{x(1+b/x)}^{x+b} =1^\infty form\\ \\ \\ e^{\lim_{x\rightarrow \infty}\big(x+b\big) \bigg(\dfrac {x+a}{x+b}-1\bigg)} \\ \\ e^{\lim_{x\rightarrow \infty }\big(x+a-x-b\big)}\\ \\ e^{\lim_{x\rightarrow \infty }\big(a-b\big)} \\ \\ e^{a-b}$