غاية بسيطة

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$\lim_{x\ to0} (\frac{\sqrt{2}^{x}-3^{x}}{\sqrt{5}^{x}-4^{x}})$

$=\lim_{x\to0} (\frac{\sqrt{2}^{x}-1-(3^{x}-1)}{\sqrt{5}^{x}-1-(4^{x}-1)})$

$\lim_{x\to0}(\frac{\frac{\sqrt{2}^{x}-1}{x}-\frac{3^{x}-1}{x}}{\frac{\sqrt{5}^{x}-1}{x}-\frac{4^{x}-1}{x}})=$

$=\frac{\ln\sqrt{2}-\ln3}{\ln\sqrt{5}-\ln4}$