Evaluate the following Limits: _ x 0 [a^x+b^x+c^x-3 x ] - Vidyadip

Question

Evaluate the following Limits:

$\lim _{x \rightarrow 0}\left[\frac{a^x+b^x+c^x-3}{x}\right]$

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Answer

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\mathrm{a}^x+\mathrm{b}^x+\mathrm{c}^x-3}{x} \\ & =\lim _{x \rightarrow 0} \frac{\left(\mathrm{a}^x-1\right)+\left(\mathrm{b}^x-1\right)+\left(\mathrm{c}^x-1\right)}{x} \\ & =\lim _{x \rightarrow 0}\left(\frac{\mathrm{a}^x-1}{x}+\frac{\mathrm{b}^x-1}{x}+\frac{\mathrm{c}^x-1}{x}\right) \\ & =\lim _{x \rightarrow 0}\left(\frac{\mathrm{a}^x-1}{x}\right)+\lim _{x \rightarrow 0}\left(\frac{\mathrm{b}^x-1}{x}\right)+\lim _{x \rightarrow 0}\left(\frac{\mathrm{c}^x-1}{x}\right) \\ & =\log \mathrm{a}+\log \mathrm{b}+\log \mathrm{c} \\ & =\log \left(\lim _{x \rightarrow 0} \frac{\mathrm{a}^x-1}{x}=\log \mathrm{a}\right] \\ & \quad \text { }\end{aligned}$

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