[tex]\displaystyle \lim_{x\rightarrow \infty }\frac{\sqrt[4]{x^{3}}+1}{\sqrt{x}+\sqrt[4]{x}+1}= \lim_{x\rightarrow \infty }\frac{\sqrt[4]{x}+\frac{1}{\sqrt[4]{x^2}}}{1+\frac{1}{\sqrt[4]{x}}+\frac{1}{\sqrt[4]{x^2}}}=\lim_{x\rightarrow \infty }\sqrt[4]{x}= \infty[/tex]