A) \[0\]
B) \[2\]
C) \[-1\]
D) \[-2\]
Correct Answer: C
Solution :
| \[\Delta (x)=\left| \begin{matrix} {{e}^{x}} \\ \cos x \\ \end{matrix}\begin{matrix} \sin x \\ \,\,\,\,In\,(1+{{x}^{2}}) \\ \end{matrix} \right|\] |
| \[{{e}^{x}}\,\,In\,\,(1+{{x}^{2}})-\sin x\,\cos x\] |
| So, \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\Delta (x)}{x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}In\,(1+{{x}^{2}})-\sin x\cos x}{x}\] |
| \[\underset{x\to 0}{\mathop{\lim }}\,\,\,x{{e}^{x}}\left\{ \frac{In\,(1+{{x}^{2}})}{{{x}^{2}}} \right\}-\underset{x\to 0}{\mathop{\lim }}\,\,\left( \frac{\sin x}{x} \right)\cos x\] |
| \[=0\times 1\times 1-1\times 1=-1\] |
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