\(\lim\limits _{x\to\infty} \left(\left(\frac x {x+1}\right)^a+ sin\left(\frac{1}{x}\right)\right)^x\)
\(= Exp\left[\lim\limits_{x \to\infty}\left(\left(\frac{x}{x +1}\right)^a + sin\left(\frac1x\right)-1\right) x\right]\)
\(= Exp\left[\lim\limits_{x \to\infty} \left(\frac{x^{a+1} -(x +1)^a x}{(x +1)^a} + x\,sin\left(\frac 1 x\right)\right)\right]\)
\(= Exp\left[\lim\limits_{x \to\infty}\left(\frac{x^{a+1}- (x^{a +1}+ ax^a + ....+x)}{x^a \left(1+\frac1x\right)^a}+x\left(\frac1x-\frac{1}{3!x^3}+...\right)\right)\right]\)
\(= Exp\,[a+1]= e^{a+1}\) (By taking limit)
