Q88 / 180 · KCET 2016 · Mathematics
\( \lim _{x \rightarrow 0} \frac{x e^{x}-\sin x}{x} \) is equal to
- A\( 13 \)
- B\( 1 \)
- Correct answerC\( 00 \)
- D\( 12 \)
Solution
Given that
\[ \begin{array}{l} \lim _{x \rightarrow 0} \frac{x e^{x}-\sin x}{x}=\lim _{x \rightarrow 0}\left(\frac{x e^{x}}{x}-\frac{\sin x}{x}\right) \\ =\lim _{x \rightarrow 0} \frac{x e^{x}}{x}-\lim _{x \rightarrow 0} \frac{\sin x}{x} \\ =1-1=0 \end{array} \]
\[ \begin{array}{l} \lim _{x \rightarrow 0} \frac{x e^{x}-\sin x}{x}=\lim _{x \rightarrow 0}\left(\frac{x e^{x}}{x}-\frac{\sin x}{x}\right) \\ =\lim _{x \rightarrow 0} \frac{x e^{x}}{x}-\lim _{x \rightarrow 0} \frac{\sin x}{x} \\ =1-1=0 \end{array} \]
Hence, the correct answer is option: 'C'.