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)
\(= 0 + \lim\limits_{x\to 0 ^+}\)
\(= \lim\limits_{h\to 0}- (0 + h)\)
\(= -0\)
\(= 0\)
