A) \[1\]
B) \[ab\]
C) \[{{e}^{ab}}\]
D) \[{{e}^{b/a}}\]
Correct Answer: C
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,{{(\cos x+a\sin bx)}^{1/x}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,{{(1+\cos x+a\sin bx-1)}^{1/x}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+a\sin bx-2{{\sin }^{2}}\frac{x}{2} \right)}^{1/x}}\] \[={{e}^{\underset{x\to 0}{\mathop{\lim }}\,\frac{a\sin bx-2{{\sin }^{2}}\frac{x}{2}}{x}}}\] \[={{e}^{\left[ \underset{x\to 0}{\mathop{\lim }}\,b\frac{a\sin bx}{bx}2\cdot \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\sin x/2}{x/2} \right)\left( \frac{1}{2}\sin \frac{x}{2} \right) \right]}}\] \[={{e}^{(ab-2\cdot 1\cdot 0)}}={{e}^{ab}}\]
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