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Question-137016




Question Number 137016 by PGeeman last updated on 28/Mar/21
Answered by Dwaipayan Shikari last updated on 28/Mar/21
lim_(x→0) a^x ∼1+xlog(a)   sin(x)∼x ;  log(1+x)∼x  lim_(x→0) (((4^x −1)^3 )/(sin((x/p))log(1+(x^2 /3))))=(((1+xlog(4)−1)^3 )/((x/p).(x^2 /3)))=3plog^3 (4)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{a}^{{x}} \sim\mathrm{1}+{xlog}\left({a}\right)\:\:\:{sin}\left({x}\right)\sim{x}\:;\:\:{log}\left(\mathrm{1}+{x}\right)\sim{x} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{4}^{{x}} −\mathrm{1}\right)^{\mathrm{3}} }{{sin}\left(\frac{{x}}{{p}}\right){log}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\right)}=\frac{\left(\mathrm{1}+{xlog}\left(\mathrm{4}\right)−\mathrm{1}\right)^{\mathrm{3}} }{\frac{{x}}{{p}}.\frac{{x}^{\mathrm{2}} }{\mathrm{3}}}=\mathrm{3}{plog}^{\mathrm{3}} \left(\mathrm{4}\right) \\ $$$$ \\ $$

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