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Q. $ \lim_{x \to {0}} \frac {a^x-1} { \sqrt {1+x}-1} $ is equal to

Solution:

$\lim_{x\to0} \frac{a^{x} - 1}{\sqrt{1 +x} -1}$
= $\lim_{x\to0} \frac{a^{x} -1}{\sqrt{1 +x } -1}. \frac{\sqrt{1+x} +1}{\sqrt{1+x} +1}$
= $\lim_{x\to0} \frac{a^{x } -1}{x} .\left(\sqrt{1+x} +1\right) = 2\left(\log_{e}a\right)$