lim-x-x-2-1-x-x-1-x-x-3-1-x-x-1-x-

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

\lim_{x \to \infty} \frac{{(x + 2)}^{1 / x} - x^{1 / x}}{{(x + 3)}^{1 / x} - x^{1 / x}} = ?

Answered by Frix last updated on 22/Jan/23

= \lim_{t \to 0^{+}} \frac{{(2 t + 1)}^{t} - 1}{{(3 t + 1)}^{t} - 1} =

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

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