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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
Question
Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{5^x+3^x-2^x-1}{x}\right]$
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Answer

$ \lim _{x \rightarrow 0} \frac{5^x+3^x-2^x-1}{x}$
$=\lim _{x \rightarrow 0} \frac{\left(5^x-1\right)+\left(3^x-2^x\right)}{x}$
$=\lim _{x \rightarrow 0} \frac{\left(5^x-1\right)+\left(3^x-1\right)-\left(2^x-1\right)}{x}$
$=\lim _{x \rightarrow 0}\left(\frac{5^x-1}{x}+\frac{3^x-1}{x}-\frac{2^x-1}{x}\right)$
$=\lim _{x \rightarrow 0}\left(\frac{5^x-1}{x}\right)+\lim _{x \rightarrow 0}\left(\frac{3^x-1}{x}\right)-\lim _{x \rightarrow 0}\left(\frac{2^x-1}{x}\right)$
$=\log 5+\log 3-\log 2 \quad \ldots\left[\lim _{x \rightarrow 0} \frac{a^x-1}{x}=\operatorname{loga}\right]$
$=\log \frac{5 \times 3}{2}$
$=\log \frac{15}{2} $

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