$$\lim_{x\rightarrow \infty}\frac{(1+x)^p}{1+x^p}=\lim_{x\rightarrow \infty}\frac{\left(x\left(\frac{1}{x}+1\right)\right)^p}{x^p\left(\frac{1}{x^p}+1\right)}=\lim_{x\rightarrow \infty}\frac{x^p\left(\frac{1}{x}+1\right)^p}{x^p\left(\frac{1}{x^p}+1\right)}=\lim_{x\rightarrow \infty}\frac{\left(\frac{1}{x}+1\right)^p}{\left(\frac{1}{x^p}+1\right)}=1$$
Da $$\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow \infty} f(x)}{\lim_{x\rightarrow \infty}g(x)}$$ $$\lim_{x\rightarrow\infty} \left(\frac{1}{x}+1\right)^p=\left(\lim_{x\rightarrow\infty} \left(\frac{1}{x}+1\right)\right)^p$$ und $$\lim_{x\rightarrow \infty}\frac{1}{x^n}=0, \forall n>0$$