求极限 $$\bex \lim_{x\to +\infty} \sex{\sqrt[6]{x^6+x^5}-\sqrt[6]{x^6-x^5}}. \eex$$

 

解答: $$\beex \bea \mbox{原极限}&=\lim_{x\to+\infty}x\sex{\sqrt[6]{1+\frac{1}{x}}-\sqrt[6]{1-\frac{1}{x}}}\\ &=\lim_{t\to 0}\frac{\sqrt[6]{1+t}-\sqrt[6]{1-t}}{t}\quad\sex{\frac{1}{x}\lra t}\\ &=\lim_{t\to 0}\frac{\frac{1}{6}(1+\xi_t)^{-\frac{5}{6}}\cdot 2t}{t} \quad\sex{f(t)=\sqrt[6]{1+t}\ra f(t)-f(-t)=f'(\xi_t)\cdot 2t}\\ &=\frac{1}{3}. \eea \eeex$$