Evaluate $\underset{x→∞}{\lim}\frac{\sqrt{x^2+1}-\sqrt[3]{x^2+1}}{\sqrt[4]{x^4+1}-\sqrt[5]{x^4+1}}$.

1

$\underset{x→∞}{\lim}\frac{\sqrt{x^2+1}-\sqrt[3]{x^2+1}}{\sqrt[4]{x^4+1}-\sqrt[5]{x^4+1}}(\frac{∞-∞}{∞-∞})$

Divide N and D by the highest power of x i.e. by x.

$\underset{x→∞}{\lim}\frac{\sqrt{x^2+1}-\sqrt[3]{x^2+1}}{\sqrt[4]{x^4+1}-\sqrt[5]{x^4+1}}=\underset{x→∞}{\lim}\frac{\sqrt{1+1/x^2}-\sqrt[3]{\frac{1}{x}+1/x^3}}{\sqrt[4]{1+\frac{1}{x^4}}-\sqrt[5]{\frac{1}{x}+\frac{1}{x^5}}}=\frac{1-0}{1-0}=1$