InterviewSolution
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InterviewSolution
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Lim x→∞((x/x+1)a + sin(1/x))x is equal to\(\lim\limits _{x\to\infty} \left(\left(\frac x {x+1}\right)^a+ sin\left(\frac{1}{x}\right)\right)^x\) |
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Answer» \(\lim\limits _{x\to\infty} \left(\left(\frac x {x+1}\right)^a+ sin\left(\frac{1}{x}\right)\right)^x\) \(= Exp\left[\lim\limits_{x \to\infty}\left(\left(\frac{x}{x +1}\right)^a + sin\left(\frac1x\right)-1\right) x\right]\) \(= Exp\left[\lim\limits_{x \to\infty} \left(\frac{x^{a+1} -(x +1)^a x}{(x +1)^a} + x\,sin\left(\frac 1 x\right)\right)\right]\) \(= Exp\left[\lim\limits_{x \to\infty}\left(\frac{x^{a+1}- (x^{a +1}+ ax^a + ....+x)}{x^a \left(1+\frac1x\right)^a}+x\left(\frac1x-\frac{1}{3!x^3}+...\right)\right)\right]\) \(= Exp\,[a+1]= e^{a+1}\) (By taking limit) |
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