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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{9^x-5^x}{4^x-1}\right]$
✓

Answer

$\lim _{x \rightarrow 0} \frac{9^x-5^x}{4^x-1}  =\lim _{x \rightarrow 0} \frac{9^x-1+1-5^x}{4^x-1}$
$ =\lim _{x \rightarrow 0} \frac{\left(9^x-1\right)-\left(5^x-1\right)}{4^x-1}$
$ =\lim _{x \rightarrow 0} \frac{\frac{\left(9^x-1\right)-\left(5^x-1\right)}{x}}{\frac{\left(4^x-1\right)}{x}} $
$[\because x \rightarrow 0, x \neq 0]$
$=\lim _{x \rightarrow 0} \frac{\left(\frac{9^x-1}{x}\right)-\left(\frac{5^x-1}{x}\right)}{\left(\frac{4^x-1}{x}\right)}$
$=\frac{\lim _{x \rightarrow 0} \frac{9^x-1}{x}-\lim _{x \rightarrow 0} \frac{5^x-1}{x}}{\lim _{x \rightarrow 0} \frac{4^x-1}{x}}$
$=\frac{\log 9-\log 5}{\log 4} \ldots\left[\because \lim _{x \rightarrow 0} \frac{a^x-1}{x}=\log a\right]$
$=\frac{1}{(\log 4)} \log \left(\frac{9}{5}\right) $

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