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Question Number 185405 by greougoury555 last updated on 21/Jan/23

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{{(x+2)}^{1/x}-x^{1/x}}{{(x+3)}^{1/x}-x^{1/x}}=?$$

Answered by Frix last updated on 22/Jan/23

$$=\underset{t\to 0^{+}}{\lim }\frac{{(2t+1)}^t-1}{{(3t+1)}^t-1}=$$$$=\underset{t\to 0^{+}}{\lim }\frac{\frac{d^2[{(2t+1)}^t-1]}{{dt}^2}}{\frac{d^2[{(3t+1)}^t-1]}{{dt}^2}}=\frac{4}{6}=\frac{2}{3}$$

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