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\( \lim _{x \rightarrow 0} \frac{(12)^{x}-(3)^{x}-(4)^{2}+1}{x \sin x} \) |
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Answer» \(\lim\limits_{x\to 0}\frac{12^x - 3^x - 4^x + 1}{x\, sinx} \) \(\left(\frac 00-case\right)\) \(= \lim\limits_{x\to0} \frac{12^x log 12 - 3^x log3 -4^xlog4 }{x\, cos x + sinx}\) \(\left(\frac 00-case\right)\) (By using D.L.H. Rule) \(= \lim\limits_{x\to0} \frac{12^x (log 12)^2 - 3^x (log3)^2 -4^x(log)^2}{-x\, sin x + 2cosx}\) (By using D.L.H. Rule) \(= \frac{(log12)^2 - (log3)^2-(log4)^2}{2}\) \(= \frac{(log 3 + log 4)^2-(log3)^2 - (log4)^2}{2}\) \((\because log12 = log(3 \times 4) = log 3 + log 4)\) \(= \frac{2log3 \,log4}{2}\) \(= log3 \,log4\) \(\therefore \lim\limits _{x\to 0} \frac{12^x - 3^x - 4^x+1} {x\, sinx}= log3\,log4 \) |
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