Question

lim x tends to 0((a^x-1) /((√1+x)-1)

Answer 1

$\lim_{x \to 0} \frac{ {a}^{x} - 1}{ \sqrt{1 + x} - 1} \\ = \lim_{x \to 0} \frac{ ({a}^{x} - 1)( \sqrt{1 + x} + 1)}{ (\sqrt{1 + x} - 1)( \sqrt{1 + x} + 1)} \\ = \lim_{x \to 0} \frac{ ({a}^{x} - 1)( \sqrt{1 + x} + 1)}{ ((1 + x) - 1)} \\ = \lim_{x \to 0} \frac{ ({a}^{x} - 1)( \sqrt{1 + x} + 1)}{ x} \\ = \lim_{x \to 0} \frac{ ({a}^{x} - 1)}{ x} \times \lim_{x \to 0} ( \sqrt{1 + x} + 1) \\ = ln(a) \times ( \sqrt{1 + 0} + 1) \\ = 2 \: ln(a) = ln( {a}^{2} )$