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\( \lim _{x \rightarrow 0^{+}}((x \cos x)+(x \log x)) \) equals(1) 1(2) 2(3) 3(4) 0 |
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Answer» Correct option is (4) 0 \(\lim\limits_{x\to 0^+} (x\, cos\, x + x\,log\, x)\) \(= \lim\limits_{x\to 0^+} x\, cosx + \lim\limits _{x \to 0^+} x\, log x\) \(= \lim\limits_{h\to 0}(0 + h)\, cos(0 + h) + \lim\limits_{x \to 0^+} \frac{log x}{\frac1x}\) \(\left(\frac{\infty}{\infty}- case\right)\) \(= 0 \times cos0 + \lim\limits_{x\to 0^+} \cfrac{\frac1x}{\frac{-1}{x^2}}\) (By using D.L.H. Rule) \(= 0 + \lim\limits_{x\to 0 ^+}\) \(= \lim\limits_{h\to 0}- (0 + h)\) \(= -0\) \(= 0\) |
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