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Expansions to Remove Indeterminate Form

lim x → 0ax+b...

Question

${lim}_{x \to 0} \frac{a^{x} + b^{x} + c^{x} - 3}{x}$

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Solution

${lim}_{x \to 0} \frac{a^{x} + b^{x} + c^{x} - 3}{x}$

$= {lim}_{x \to 0} \frac{(a^{x} - 1) + (b^{x} - 1) + (c^{x} - 1)}{x}$

$= {lim}_{x \to 0} \frac{a^{x} - 1}{x} + {lim}_{x \to 0} \frac{b^{x} - 1}{x} + {lim}_{x \to 0} \frac{c^{x} - 1}{x}$

$= l o g a + l o g b + l o g c$

$= l o g (a b c)$

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