Evaluate the following limits: _ x 0 (a^x+b^x+c^x-3 x )

Question

Evaluate the following limits: $\lim _{x \rightarrow 0}\left(\frac{a^x+b^x+c^x-3}{\sin x}\right)$

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Answer

$\lim _{x \rightarrow 0} \frac{a^x+b^x+c^x-3}{\sin x}$
$=\lim _{x \rightarrow 0} \frac{\left(a^x-1\right)+\left(b^x-1\right)+\left(c^x-1\right)}{\sin x}$
$=\lim _{x \rightarrow 0} \frac{\left(\mathrm{a}^x-1\right)+\left(\mathrm{b}^x-1\right)+\left(\mathrm{c}^x-1\right)}{x} [$ Divide numerator and denominator by $x \quad \because x \rightarrow 0, x \neq 0]$
$=\lim _{x \rightarrow 0} \frac{\left(\frac{a^x-1}{x}\right)+\left(\frac{b^x-1}{x}\right)+\left(\frac{c^x-1}{x}\right)}{\frac{\sin x}{x}}$
$=\frac{\left(\lim _{x \rightarrow 0} \frac{\mathrm{a}^x-1}{x}\right)+\left(\lim _{x \rightarrow 0} \frac{\mathrm{b}^x-1}{x}\right)+\left(\lim _{x \rightarrow 0} \frac{\mathrm{c}^x-1}{x}\right)}{\left(\lim _{x \rightarrow 0} \frac{\sin x}{x}\right)}$
$=\frac{\log \mathrm{a}+\log \mathrm{b}+\log \mathrm{c}}{1} \ldots .\left[\because \lim _{x \rightarrow 0} \frac{\mathrm{a}^x-1}{x}=\log \mathrm{a}\right]$
$=\log (a b c)$

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