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Question Number 132537 by SLVR last updated on 15/Feb/21

$$Iff\left(x\right)=8x^3+3xthen{lim}_{x\to \mathrm{\infty }}\frac{x^{1/3}}{f^{-1}\left(8x\right)-f^{-1}\left(x\right)}is$$

Commented by SLVR last updated on 15/Feb/21

$$cananyonehelpme...please$$

Answered by Ñï= last updated on 16/Feb/21

$$x=8t^3+3t$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{f^{-1}\left(x\right)}{x^{1/3}}=\underset{t\to \mathrm{\infty }}{\lim }\frac{f^{-1}(8t^3+3t)}{{(8t^3+3t)}^{1/3}}=\underset{t\to \mathrm{\infty }}{\lim }\frac{t}{{(8t^3+3t)}^{1/3}}=\frac{1}{2}$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{f^{-1}\left(8x\right)}{x^{1/3}}=2\underset{x\to \mathrm{\infty }}{\lim }\frac{f^{-1}\left(8x\right)}{{\left(8x\right)}^{1/3}}=1$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{x^{1/3}}{f^{-1}\left(8x\right)-f^{-1}\left(x\right)}=\frac{1}{1-\frac{1}{2}}=2$$

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